{"id":12650,"date":"2019-12-18T16:26:35","date_gmt":"2019-12-18T10:56:35","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=12650"},"modified":"2021-07-02T15:35:56","modified_gmt":"2021-07-02T10:05:56","slug":"1st-puc-maths-question-bank-chapter-3","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/1st-puc-maths-question-bank-chapter-3\/","title":{"rendered":"1st PUC Maths Question Bank Chapter 3 Trigonometric Functions"},"content":{"rendered":"

Students can Download Maths Chapter 3 Trigonometric Functions Questions and Answers, Notes Pdf, 1st PUC Maths Question Bank with Answers<\/a> helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n

Karnataka 1st PUC Maths Question Bank Chapter 3 Trigonometric Functions<\/h2>\n

Measurement of Angles:<\/strong><\/p>\n

Question 1.
\nWhat is the meaning of trigonometry?
\nAnswer :
\nThe literal meaning of the word trigonometry is the science of measuring the sides and the angles of triangles.<\/p>\n

Question 2.
\nDefine an angle.
\nAnswer :
\nAngle is a measure of rotation of a given ray about its initial point.
\n\"1st<\/p>\n

Question 3.
\nDefine a degree measure.
\nAnswer :
\nIf a rotation from the initial side to terminal side is \\( \\left(\\frac{1}{360}\\right)^{t h} \\) of a revolution, the angle is said to have a measure of one degree, written as 1\u00b0.<\/p>\n

Keen eye:
\nA degree is divided into 60 minutes and a minute is divided into 60 seconds. One sixtieth of a degree is called a minute, written as 1′ and one sixtieth of a minute is called second, written as 1″
\n\u2234 1 \u00b0 – 60′
\n1’= 60″<\/p>\n

\"KSEEB<\/p>\n

Question 4.
\nDefine a radian measure.
\nAnswer :
\nAngle subtended at the centre by an arc of length 1 unit in a unit circle is said to have a measure of 1 radian.<\/p>\n

Keen eye:
\n\\(1 \\mathrm{rad}=\\frac{180^{\\circ}}{\\pi}=57^{\\circ} 16^{\\prime} \\)<\/p>\n\n\n\n\n
Degree<\/td>\n30\u00b0<\/td>\n45\u00b0<\/td>\n60\u00b0<\/td>\n90\u00b0<\/td>\n180\u00b0<\/td>\n270\u00b0<\/td>\n360\u00b0<\/td>\n<\/tr>\n
Radian<\/td>\n\\(\\frac{\\pi}{6}\\)<\/td>\n\\(\\frac{\\pi}{4}\\)<\/td>\n\\(\\frac{\\pi}{3}\\)<\/td>\n\\(\\frac{\\pi}{2}\\)<\/td>\n\u03c0<\/td>\n\\(\\frac{3 \\pi}{2}\\)<\/td>\n2\u03c0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Question 5.
\nConvert the following into radian measure,
\n(i) 25\u00b0
\n(ii) 240\u00b0
\n(iii) 520\u00b0
\n(iv) 225\u00b0
\n(v) 150\u00b0
\n(vi) 330\u00b0
\n(vii) – 47\u00b0 30′
\n(viii) 40\u00b0 20′
\nAnswer:
\n\"1st<\/p>\n

Question 6.
\nConvert the following into degree measure.
\n(i) 6
\n(ii) \\( \\frac{11}{16}\\)
\n(iii) -4
\n(iv) \\( \\frac{5 \\pi}{3} \\)
\n(v) \\( \\frac{7 \\pi}{6} \\)
\nAnswer:
\n\"1st
\n\"1st
\n\"1st<\/p>\n

Question 7.
\nA wheel makes 360\u00b0 revolutions in one minute. Through how many radians does it turn in one second?
\nAnswer :
\nNumber of revolutions in one minute = 360
\nNumber of revolutions in one second = \\( \\frac{360}{60}=6\u00a0 \\)
\nBut, in one revolutions angle traced = 360\u00b0
\n= 2\u03c0 rad
\n\u21d2 Angle traced in 6 revolutions
\n= 6 x 2\u03c0= 12\u03c0 radians
\nAngle traced in one second = 12\u03c0 radians<\/p>\n

\"KSEEB<\/p>\n

Question 8.
\nFind the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm \\(\\left(\\text { use } \\pi=\\frac{22}{7}\\right)\\)
\nAnswer :
\n\"1st<\/p>\n

Question 9.
\nFind the radius of the circle in which a central angle of 60\u00b0 intercepts an arc of length \\(37.4 \\mathrm{cm}\\left(\\text { use } \\pi=\\frac{22}{7}\\right)\\)
\nAnswer :
\n\"1st<\/p>\n

Question 10.
\nIn a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
\nAnswer :
\nGiven, diameter AB = 40 cm
\n\"1st
\n\u2234 Radius OB = 20 cm
\nChord BC = 20 cm
\n\u2234\u00a0 \u0394 OBC is an equilateral triangle
\n\"1st<\/p>\n

Question 11.
\nThe minute hand of a watch is 1.5 cm long. How far does its tip move in 40 minute?
\n(use \u03c0 = 3.14).
\nAnswer :
\nIn 60 minutes, the minute hand of a watch completes one revolution.
\n\u2234 In 40 minutes, the minute hand turns through\u00a0 of a revolution
\n\"1st<\/p>\n

Question 12.
\nIf the arcs of the same lengths in two circles subtend angles 65\u00b0 and 110\u00b0 at the centre, find the ratio of their radii.
\nAnswer :
\n\"1st<\/p>\n

Question 13.
\nIf in two circles, arcs of the same length subtend angles 60\u00b0 and 75\u00b0 at the centre, find the ratio of their radii.
\nAnswer :
\nLet r1<\/sub> and r2<\/sub> be the radii of two circles
\n\\( \\text { Given } \\quad \\theta_{1}=60^{\\circ}=\\frac{\\pi}{3} \\mathrm{rad}\\)
\n\"1st<\/p>\n

Question 14.
\nFind the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
\n(i) 10 cm
\n(ii) 15 cm
\n(iii) 21 cm
\nAnswer :
\n\"1st<\/p>\n

Trigonometric Functions:<\/p>\n

Recall:
\n\"1st<\/p>\n

\"1st<\/p>\n

\"1st<\/p>\n

\"1st<\/p>\n

Graph of trigonometric function:
\n\"1st
\n\"1st
\n\"1st<\/p>\n

Question 1.
\nIf \\(\\cos x=-\\frac{3}{5}, x\\) lies in the third quadrant, find the values of other five trigonometric functions.
\nAnswer
\n\"1st<\/p>\n

Question 2.
\nIf \\( \\cos x=-\\frac{1}{2}, x \\) lies in the third quadrant, find the values of other five trigonometric functions.
\nAnswer:
\n\"1st<\/p>\n

Question 3.
\nIf \\( \\sin x=\\frac{3}{5}, x \\) lies in the second quadrant, find the values of other five trigonometric functions.
\nAnswer:
\n\"1st
\n\"1st<\/p>\n

Question 4.
\nIf \\( \\tan x=-\\frac{5}{12}, x \\)\u00a0 lies in the second quadrant, find the values of other five trigonometric functions.
\nAnswer:
\n\"1st<\/p>\n

Question 5.
\nIf \\(\\cot x=\\frac{3}{4}, x \\) lies in third quadrant, find the values of other five trigonometric functions.
\nAnswer:
\n\"1st
\n\"1st<\/p>\n

Question 6.
\nIf \\(\\\\cot x=-\\frac{5}{12}, x lies\\) in second quadrant, find the values of other five trigonometric functions.
\nAnswer:
\n\"1st
\n\\(\\cot x=-\\frac{5}{12}\\)<\/p>\n

Question 7.
\nIf \\(\\sec x=\\frac{13}{5}, x\\) lies in fourth quadrant, find the values of other five trigonometric functions.
\nAnswer:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
Measure of angle<\/strong><\/td>\nsin<\/strong><\/td>\ncos<\/strong><\/td>\ntan<\/strong><\/td>\ncosec<\/strong><\/td>\nsec<\/strong><\/td>\ncot<\/strong><\/td>\n<\/tr>\n
-\u03b8<\/td>\n– sin\u03b8<\/td>\ncos\u03b8<\/td>\n-tan\u03b8<\/td>\n– cosec\u03b8<\/td>\nsec\u03b8<\/td>\n– cot\u03b8<\/td>\n<\/tr>\n
\\(\\frac{\\pi}{2}-\\theta \\)<\/td>\ncos\u03b8<\/td>\n-sin\u03b8<\/td>\ncot \u03b8<\/td>\nsec\u03b8<\/td>\ncosec\u03b8<\/td>\ntan\u03b8<\/td>\n<\/tr>\n
\\(\\frac{\\pi}{2}+\\theta \\)<\/td>\ncos\u03b8<\/td>\n– sin\u03b8<\/td>\n-cot\u03b8<\/td>\n– sec\u03b8<\/td>\ncosec\u03b8<\/td>\n-tan\u03b8<\/td>\n<\/tr>\n
\u03c0-\u03b8<\/td>\nsin\u03b8<\/td>\n-cos\u03b8<\/td>\n-tan\u03b8<\/td>\ncosec \u03b8<\/td>\n-sec\u03b8<\/td>\n-cot\u03b8<\/td>\n<\/tr>\n
\u03c0+\u03b8<\/td>\n– sin\u03b8<\/td>\n-cos\u03b8<\/td>\ntan\u03b8<\/td>\n-coscc\u03b8<\/td>\n– sec\u03b8<\/td>\ncot\u03b8<\/td>\n<\/tr>\n
\\(\\frac{3 \\pi}{2}-\\theta\\)<\/td>\n– cos\u03b8<\/td>\n-sin\u03b8<\/td>\ncot\u03b8<\/td>\n– sec\u03b8<\/td>\n– cosec\u03b8<\/td>\ntan \u03b8<\/td>\n<\/tr>\n
\\(\\frac{3 \\pi}{2}+\\theta\\)<\/td>\n-cos\u03b8<\/td>\nsin\u03b8<\/td>\n-cot\u03b8<\/td>\n– sec\u03b8<\/td>\ncosec\u03b8<\/td>\ntan\u03b8<\/td>\n<\/tr>\n
2\u03c0-\u03b8<\/td>\n-sin\u03b8<\/td>\ncos\u03b8<\/td>\n-tan\u03b8<\/td>\n– coscc\u03b8<\/td>\nsec\u03b8<\/td>\n-cot\u03b8<\/td>\n<\/tr>\n
2\u03c0+\u03b8<\/td>\nsin\u03b8<\/td>\ncos \u03b8<\/td>\ntan\u03b8<\/td>\ncosec\u03b8<\/td>\nsec\u03b8<\/td>\ncos\u03b8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Question 8.
\nFind the value of
\n\\( \\sin \\frac{31 \\pi}{3} \\)
\nAnswer:
\nWe know that values of sinx repeats after an interval of 2\u03c0.
\n\\( \\sin \\frac{31 \\pi}{3}=\\sin \\left(10 \\pi+\\frac{\\pi}{3}\\right)=\\sin \\frac{\\pi}{3}=\\frac{\\sqrt{3}}{2}\\)<\/p>\n

\"KSEEB<\/p>\n

Question 9.
\nFind the value of sin 765\u00b0.
\nAnswer:
\n\\( \\sin 765^{\\circ}=\\sin \\left(720^{\\circ}+45^{\\circ}\\right)=\\sin 45^{\\circ}=\\frac{1}{\\sqrt{2}}\\)<\/p>\n

Question 10.
\nFind the value of
\n\\( \\sin \\left(-\\frac{11 \\pi}{3}\\right) \\)
\nAnswer:
\n\\(\\begin{aligned} \\sin \\left(-\\frac{11 \\pi}{3}\\right) &=-\\sin \\left(\\frac{11 \\pi}{3}\\right)=-\\sin \\left(4 \\pi-\\frac{\\pi}{3}\\right) \\\\ &=\\sin \\left(\\frac{\\pi}{3}\\right)=\\frac{\\sqrt{3}}{2} \\end{aligned}\\)<\/p>\n

Question 11.
\nFind the value of cos (-1710\u00b0)
\nAnswer :
\ncos (1710\u00b0) = cos (1710\u00b0)
\n\\( \\begin{array}{l}{=\\cos \\left(1800-90^{\\circ}\\right)=\\cos \\left(10 \\pi-\\frac{\\pi}{2}\\right)} \\\\ {=\\cos \\frac{\\pi}{2}=0}\\end{array}\\)<\/p>\n

Question 12.
\nFind the value of cosec (-1410\u00b0)
\nAnswer :
\ncosec (- 1410\u00b0)
\n= – cosec (1410\u00b0)
\n= – cosec[4(360\u00b0) – 30\u00b0]
\n= – cosec (- 30\u00b0)
\n= cosec 30\u00b0 = 2<\/p>\n

Question 13.
\nFind the value of
\n\\(\u00a0 \\tan \\frac{19 \\pi}{3}\\)
\nAnswer:
\n\\(\\tan \\frac{19 \\pi}{3}=\\tan \\left(6 \\pi+\\frac{\\pi}{3}\\right)=\\tan \\frac{\\pi}{3}=\\sqrt{3} \\)<\/p>\n

Question 14.
\nFind the value of
\n\\( \\cot \\left(-\\frac{15 \\pi}{4}\\right)\\)
\nAnswer:
\n\\(\\begin{aligned} \\cot \\left(-\\frac{15 \\pi}{4}\\right) &=-\\cot \\left(\\frac{15 \\pi}{4}\\right)=-\\cot \\left(4 \\pi-\\frac{\\pi}{4}\\right) \\\\ &=\\cot \\frac{\\pi}{4}=1 \\end{aligned}\\)<\/p>\n

Question 15.
\nDerive cos(x + y) = cos x cos y – sin x sin y , using unit circle concept.
\n\"1st
\nAnswer:
\n\"1st
\n\"1st
\n<\/span><\/p>\n

Question 16.
\nProve that:
\ncos(x-y) = cosxcosy+ sinxsiny.
\nAnswer :
\nWe have cos(x + y) = cos xcos y – sin xsin y
\nReplace y by – y, we get
\ncos[x + (-y)] = cos xcos(-y) – sin xsin(-y)
\ncos(x – y) = cos x cos y + sin x sin y<\/p>\n

Question 17.
\nProve that
\n\\( \\cos \\left(\\frac{\\pi}{2}-x\\right)=\\sin x \\)
\nAnswer:
\n\"1st<\/p>\n

Question 18.
\nProve that
\n\\( \\sin \\left(\\frac{\\pi}{2}-x\\right)=\\cos x\\)
\nAnswer:
\n\"1st<\/p>\n

Question 19.
\nProve that sin(x + y) = sinx cosy + cosx siny .
\nAnswer :
\n\"1st
\n\"1st<\/p>\n

Question 20.
\nProve that sin( x – y) = sin x cos y – cos x sin y.
\nAnswer :
\nsin(x – y) = sin[x + (-y)]
\n= sin x cos(-y) + cos x\u00a0 sin(-y)
\n= sin x cos y-cos x sin y .<\/p>\n

\"KSEEB<\/p>\n

Question 21.
\nProve the following
\n(i) \\(\\cos \\left(\\frac{\\pi}{2}+x\\right)=-\\sin x \\)
\n(ii) \\( \\sin \\left(\\frac{\\pi}{2}+x\\right)=\\cos x \\)
\n(iii) cos(\u03c0 – x) = – cosx
\n(iv) sin(\u03c0 – x) =sinx
\n(v) cos(\u03c0 + x) = – cosx
\n(vi) sin(\u03c0 + x) = – sinx
\n(vii) cos(2\u03c0 – x) =cosx
\n(viii) sin(2\u03c0 – x) = – sin x.
\nAnswer:
\n\"1st<\/p>\n

Question 22.
\nIf none of the angles ac, y and x + y is an odd multiple of \\(\\frac{\\pi}{2} \\)then prove that
\n\\(\\tan (x \\pm y)=\\frac{\\tan x \\pm \\tan y}{1 \\mp \\tan x \\tan y} \\)
\nAnswer:
\n\"1st
\n\"1st<\/p>\n

Question 23.
\nIf none of the angles x,y and (x+y) is a multiple of \u03c0, then prove that
\n\\( \\cot (x \\pm y)=\\frac{\\cot x \\cot y \\pm 1}{\\cot y \\pm \\cot x}\\)
\nAnswer :
\nAs x, y and x+y are not the multiple of \u03c0, so Sinx, sin y and sin(x+ y) are non zero.
\n\"1st
\n\"1st<\/p>\n

Question 24.
\nProve that
\n\\(\\cos 2 x=\\left\\{\\begin{array}{l}{\\cos ^{2} x-\\sin ^{2} x} \\\\ {2 \\cos ^{2} x-1} \\\\ {1-2 \\sin ^{2} x} \\\\ {\\frac{1-\\tan ^{2} x}{1+\\tan ^{2} x}}\\end{array}\\right.\\)
\nAnswer:
\nWe have cos(x + y) = cos xcos y – sin xsin y
\nReplacing y by x, we get
\ncos(x + x) = cosx cosx-sinx sinx
\n\"1st<\/p>\n

Question 25.
\nProve that
\n\\( \\sin 2 x=\\left\\{\\begin{array}{c}{2 \\sin x \\cos x} \\\\ {\\frac{2 \\tan x}{1+\\tan ^{2} x}}\\end{array}\\right.\\)
\nAnswer:
\nWe have sin(x + y) = sin xcos y + cos xsin y
\nReplacing y by x we get
\nsin 2x = 2sinx cosx<\/p>\n

\"1st<\/p>\n

Question 26.
\nProve that
\n\\(\\tan 2 x=\\frac{2 \\tan x}{1-\\tan ^{2} x} \\)
\nAnswer:
\n\"1st<\/p>\n

\"KSEEB<\/p>\n

Question 27.
\nProve that sin3x = 3sinx – 4sin3<\/sup> x .
\nAnswer :
\nsin 3x = sin(2x + x)
\n= sin 2x cos x + cos 2x sin x
\n= 2sin xcos x cos x + (1 – 2sin2<\/sup> x)sin x
\n= 2 sin x(1- sin2<\/sup> x) + sin x – 2 sin3<\/sup> x
\n= 3sinx-4sin3<\/sup> x<\/p>\n

Question 28.
\nProve that cos3x = 4cos3<\/sup> x -3cosx .
\nAnswer :
\ncos3x = cos(2x + x)
\n= cos 2x cos x – sin 2x sin x
\n= (2 cos2<\/sup> x -1) cos x – 2 sin x cos x sin x
\n= 2cos3<\/sup>x-cosx-2cosx(1-cos2<\/sup>x)
\n= 4cos3<\/sup> x-3cosx<\/p>\n

Question 29.
\nProve that
\n\\(\\tan 3 x=\\frac{3 \\tan x-\\tan ^{3} x}{1-3 \\tan ^{2} x}\\)
\nAnswer:
\n\"1st
\n\"1st<\/p>\n

Question 30.
\nProve the following
\n(i) \\(\\cos x+\\cos y=2 \\cos \\frac{x+y}{2} \\cos \\frac{x-y}{2}\\)
\n(ii) \\( \\cos x-\\cos y=-2 \\sin \\frac{x+y}{2} \\sin \\frac{x-y}{2} \\)
\n(iii) \\( \\sin x+\\sin y=2 \\sin \\frac{x+y}{2} \\cos \\frac{x-y}{2} \\)
\n(iv) \\( \\sin x-\\sin y=2 \\cos \\frac{x+y}{2} \\sin \\frac{x-y}{2} \\)
\nAnswer:
\nWe have
\ncos(A + B) = cosAcosB-sinAsinB …(1)
\ncos(A – B) = cos AcosB + sin Asin B … (2)
\nAdding and subtracting (1) and (2) we get
\ncos( A + B) + cos( A -B) = 2 cos A cos B … (3)
\ncos( A + B) – cos (A -B) = – 2 sin A sin B … (4)
\nFurther
\nsin(A + B) = sinAcosB + cosAsinB\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 …(5)
\nsin(A-B) = sinAcosB-cosAsinB …(6)
\nAdding and subtracting (5) and (6) we get
\nsin( A + B) + sin( A – B) = 2 sin A cos B … (7)
\nsin( A + B) – sin(A – B) = 2cosA sinB … (8)
\nLet A + B = x and A-B = y
\n\"1st
\n\"1st
\nKeen eye;<\/p>\n

    \n
  • 2 cos xcos y = cos(x + y) + cos(x – y)<\/li>\n
  • -2 sin x sin y = cos(x + y) – cos(x – y)<\/li>\n
  • 2sin x cos y = sin(x + y) 4-sin(x- y)<\/li>\n
  • 2cosx siny = sin(x+ y)-sin(x-y)<\/li>\n<\/ul>\n

    \"KSEEB<\/p>\n

    Question 31.
    \nProve the following
    \n(i) \\( 3 \\sin \\frac{\\pi}{6} \\sec \\frac{\\pi}{3}-4 \\sin \\frac{5 \\pi}{6} \\cot \\frac{\\pi}{4}=1\\)
    \n(ii) \\( \\sin ^{2} \\frac{\\pi}{6}+\\cos ^{2} \\frac{\\pi}{3}-\\tan ^{2} \\frac{\\pi}{4}=-\\frac{1}{2}\\)
    \n(iii) \\( 2 \\sin ^{2} \\frac{\\pi}{6}+\\csc ^{2} \\frac{7 \\pi}{6} \\cos ^{2} \\frac{\\pi}{3}=\\frac{3}{2} \\)
    \n(iv) \\( 2 \\sin ^{2} \\frac{3 \\pi}{4}+2 \\cos ^{2} \\frac{\\pi}{4}+2 \\sec ^{2} \\frac{\\pi}{3}=10 \\)
    \n(v) \\( \\cot ^{2} \\frac{\\pi}{6}+\\csc \\frac{5 \\pi}{6}+3 \\tan ^{2} \\frac{\\pi}{6}=6 \\)
    \nAnswer:
    \n\"1st
    \n\"1st<\/p>\n

    Question 32.
    \nFind the Value of
    \n(i) sin 75\u00b0
    \n(ii) cos 75\u00b0
    \n(iii) tan 75\u00b0
    \n(iv) sin 15\u00b0
    \n(v) cos 15\u00b0
    \n(vi) tan 15\u00b0
    \n(vii) sin 105\u00b0
    \n(viii) cos 105\u00b0.
    \nAnswer:
    \n\"1st
    \n\"1st<\/p>\n

    Question 33.
    \nFind the value of \\( \\tan \\frac{13 \\pi}{12} \\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 34.
    \nProve that
    \n\\( \\frac{\\sin (x+y)}{\\sin (x-y)}=\\frac{\\tan x+\\tan y}{\\tan x-\\tan y} \\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 35.
    \nProve that
    \n\\(\\begin{array}{l}{\\cos \\left(\\frac{\\pi}{4}-x\\right) \\cos \\left(\\frac{\\pi}{4}-y\\right)-\\sin \\left(\\frac{\\pi}{4}-x\\right) \\sin \\left(\\frac{\\pi}{4}-y\\right)} \\\\ {=\\sin (x+y)}\\end{array} \\)
    \nAnswer:
    \n\\( \\begin{aligned} L H S &=\\cos \\left[\\frac{\\pi}{4}-x+\\frac{\\pi}{4}-y\\right] \\\\ &=\\cos \\left[\\frac{\\pi}{2}-(x+y)\\right] \\\\ &=\\sin (x+y)=R H S \\end{aligned} \\)<\/p>\n

    Question 36.
    \nProve that
    \nsin[(n + l)x]sin[(n + 2)x] +cos[(n+1)x]cos[(n + 2)x] = cosx
    \nAnswer :
    \nLHS = cos[(n + 2)x – (n + 1)x]
    \n= cos x = RHS .<\/p>\n

    Question 37.
    \nProve that
    \n(sin3x + sinx)sinx + (cos3x -cosx)cosx = 0
    \nAnswer :
    \nLHS = sin 3x\u00a0 sin x+sin2<\/sup> x + cos 3xcos x – cos2<\/sup> x
    \n= [cos 3x cos x + sin 3x sin x] – [cos2<\/sup> x – sin2<\/sup> x]
    \n= cos(3x – x) – cos 2x
    \n= 0
    \n= RHS<\/p>\n

    Question 38.
    \nProve that
    \n\\(\\frac{\\tan \\left(\\frac{\\pi}{4}+x\\right)}{\\tan \\left(\\frac{\\pi}{4} x\\right)}=\\left(\\frac{1+\\tan x}{1-\\tan x}\\right)^{2} \\)
    \nAnswer:
    \n\\( L H S=\\frac{\\frac{1+\\tan x}{1-\\tan x}}{\\frac{1-\\tan x}{1+\\tan x}}=\\left(\\frac{1+\\tan x}{1-\\tan x}\\right)^{2}=R H S\\)<\/p>\n

    \"KSEEB<\/p>\n

    Question 39.
    \nProve that
    \n\\( \\frac{\\cos (\\pi+x) \\cos (-x)}{\\sin (\\pi-x) \\cos \\left(\\frac{\\pi}{2}+x\\right)}=\\cot ^{2} x\\)
    \nAnswer:
    \n\\(L H S=\\frac{(-\\cos x)(\\cos x)}{(\\sin x)(-\\sin x)}=\\cot ^{2} x=R H S \\)<\/p>\n

    Question 40.
    \nProve that
    \n\\( \\begin{array}{l}{\\cos \\left(\\frac{3 \\pi}{2}+x\\right) \\cos (2 \\pi+x)} \\\\ {\\left[\\cot \\left(\\frac{3 \\pi}{2}-x\\right)+\\cot (2 \\pi+x)\\right]=1}\\end{array}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 41.
    \nProve that
    \n\\(\\cos \\left(\\frac{3 \\pi}{4}+x\\right)-\\cos \\left(\\frac{3 \\pi}{4}-x\\right)=-\\sqrt{2} \\sin x\\)
    \nAnswer:<\/p>\n

    \"1st<\/p>\n

    Question 42.
    \nProve that sin2<\/sup>6x-sin2<\/sup>4x=sin2xsinl0x.
    \nAnswer :
    \nLHS = sin(6x + 4x)sin(6x – 4x)
    \n\u2235 sin( A + B) sin(A – B)\u00a0 = sin2<\/sup> A – sin2<\/sup> B
    \nLHS = sin 10x . sin2x
    \n= RHS.<\/p>\n

    Question 43.
    \nProve that cos2<\/sup> 2x-cos2<\/sup>6x =sin4x sin8x .
    \nAnswer :
    \nLHS = (1 – sin2<\/sup> 2x) – (1 – sin2<\/sup> 6x)
    \n= sin2<\/sup> 6x – sin2<\/sup> 2x = sin(6x + 2x) sin(6x – 2x)
    \n= sin8x sin4x
    \n= RHS<\/p>\n

    Question 44.
    \nProve that
    \n\\( \\cos \\left(\\frac{\\pi}{4}+x\\right)+\\cos \\left(\\frac{\\pi}{4}-x\\right)=\\sqrt{2} \\cos x \\)
    \nAnswer:
    \n\\(\\begin{aligned} L H S &=\\cos \\frac{\\pi}{4} \\cos x-\\sin \\frac{\\pi}{4} \\sin x+\\cos \\frac{\\pi}{4} \\cos x+\\sin \\frac{\\pi}{4} \\sin x \\\\ &=2 \\cos \\frac{\\pi}{4} \\cos x=2\\left(\\frac{1}{\\sqrt{2}}\\right) \\cos x=\\sqrt{2} \\cos x \\\\ &=R H S \\end{aligned} \\)<\/p>\n

    Question 45.
    \nProve that
    \ntan3x tan2x tanx = tan3x-tan2x-tanx.
    \nAnswer :
    \nConsider: tan 3x = tan(2x + x) tan 2x + tan x
    \n\\( \\tan 3 x=\\frac{\\tan 2 x+\\tan x}{1-\\tan 2 x \\tan x} \\)
    \n\u21d2 tan 3x(1 – tan 2x tan x) = tan 2x + tan x
    \n\u21d2 tan3x-tan3x tan2x tanx = tan2x + tanx
    \n\u2234 tan 3x – tan 2x – tan x = tan 3x tan 2x tan x<\/p>\n

    Question 46.
    \nProve that
    \ncot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1.
    \nAnswer :
    \nConsider cot 3x = cot(2x + x)
    \n\\( =\\frac{\\cot 2 x \\cot x-1}{\\cot 2 x+\\cot x} \\)
    \n\u21d2 cot 3x(cot 2x+cot x) – cot 2x cot x -1
    \n\u21d2 1 = cot 2x cot x – cot 3x cot 2x – cot 3x cot x<\/p>\n

    Question 47.
    \nProve that
    \n\\(\\frac{\\cos 9 x-\\cos 5 x}{\\sin 17 x-\\sin 3 x}=-\\frac{\\sin 2 x}{\\cos 10 x}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 48.
    \nProve that
    \n\\(\\frac{\\sin 5 x+\\sin 3 x}{\\cos 5 x+\\cos 3 x}=\\tan 4 x \\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 49.
    \nProve that
    \n\\(\\frac{\\sin x+\\sin 3 x}{\\cos x+\\cos 3 x}=\\tan 2 x\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 50.
    \nProve that
    \n\\(\\frac{\\cos 7 x+\\cos 5 x}{\\sin 7 x-\\sin 5 x}=\\cot x\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 51.
    \nProve that
    \n\\(\\frac{\\sin x-\\sin y}{\\cos x+\\cos y}=\\tan \\left(\\frac{x-y}{2}\\right) \\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 52.
    \nProve that
    \n\\(\\frac{\\sin x-\\sin 3 x}{\\sin ^{2} x-\\cos ^{2} x}=2 \\sin x \\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 53.
    \nProve that
    \n\\(\\frac{\\cos 4 x+\\cos 3 x+\\cos 2 x}{\\sin 4 x+\\sin 3 x+\\sin 2 x}=\\cot 3 x \\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 54.
    \nProve that
    \n\\(\\frac{\\sin 5 x-2 \\sin 3 x+\\sin x}{\\cos 5 x-\\cos x}=\\tan x\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 55.
    \nProve that
    \n\\(\\frac{\\sin 9 x+\\sin 7 x+\\sin 5 x+\\sin 3 x}{\\cos 9 x+\\cos 7 x+\\cos 5 x+\\cos 3 x}=\\tan 6 x\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 56.
    \nProve that
    \n\\((\\cos x+\\cos y)^{2}+(\\sin x-\\sin y)^{2}=4 \\cos ^{2}\\left(\\frac{x+y}{2}\\right)\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 57.
    \nProve that
    \n\\((\\cos x-\\cos y)^{2}+(\\sin x-\\sin y)^{2}=4 \\sin ^{2}\\left(\\frac{x-y}{2}\\right)\\)
    \nAnswer:
    \n\"1st
    \n\"1st<\/p>\n

    Question 58.
    \nProve that
    \nsinx + sin3x + sin5x + sin7x
    \n= 4 cos x cos 2x sin 4x
    \nAnswer :
    \nLHS = (sin 7x + sin x) + (sin 5x + sin 3x)
    \n= 2 sin 4x cos 3x + 2 sin 4x cos x
    \n= 2 sin 4x(cos 3x + cos x)
    \n= 2 sin 4x 2 cos 2x cos x
    \n= 4cosx cos2x sin4x
    \n= RHS.<\/p>\n

    \"KSEEB<\/p>\n

    Question 59.
    \nProve that
    \n\\(\\sin 3 x+\\sin 2 x-\\sin x=4 \\sin x \\cos \\frac{x}{2} \\cos \\frac{3 x}{2} \\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 60.
    \nProve that
    \n\\(2 \\cos \\frac{\\pi}{13} \\cos \\frac{9 \\pi}{13}+\\cos \\frac{3 \\pi}{13}+\\cos \\frac{5 \\pi}{13}=0\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 61.
    \nProve that
    \n\\(\\tan 4 x=\\frac{4 \\tan x\\left(1-\\tan ^{2} x\\right)}{1-6 \\tan ^{2} x+\\tan ^{4} x}\\)
    \nAnswer:
    \n\"1st
    \n\"1st<\/p>\n

    Question 62.
    \nProve that cos4x = 1 – 8sin2<\/sup> x cos2<\/sup> x.
    \nAnswer :
    \nLHS = cos2(2x)
    \n= 1-2sin2<\/sup> 2x = 1-2(2sinx cosx)2
    \n<\/sup>= 1-8sin2<\/sup>xcos2<\/sup>x
    \n= RHS<\/p>\n

    Question 63.
    \nProve that
    \ncos6x = 32cos6<\/sup> x – 48cos4<\/sup> x + 18cos2<\/sup> x -1.
    \nAnswer :
    \nLHS = cos3(2x)
    \n= 4cos3<\/sup> 2x-3cos2x
    \n= 4(2 cos2<\/sup> x -1)3<\/sup> – 3(2 cos2<\/sup> x -1)
    \n= 4[8 cos6<\/sup> x -1 – 3(2cos2<\/sup> x)(2cos2<\/sup> x -1)] -6cos2<\/sup>x + 3
    \n= 32cos6<\/sup>x-4-24cos2<\/sup>x(2cos2<\/sup>x-1) -6cos2<\/sup>x + 3
    \n= 32 cos6<\/sup> x – 48 cos4<\/sup> x +18 cos2<\/sup> x -1
    \n= RHS.<\/p>\n

    Question 64.
    \nIf \\(\\sin x=\\frac{3}{5}, \\cos y=-\\frac{12}{13} \\)
    \nAnswer:
    \n\"1st
    \n\"1st<\/p>\n

     <\/p>\n

    Question 65.
    \nProve that
    \n\\(\\cos 2 x \\cos \\frac{x}{2}-\\cos 3 x \\cos \\frac{9 x}{2}=\\sin 5 x \\sin \\frac{5 x}{2}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 66.
    \nFind the value of\u00a0 \\( \\tan \\frac{\\pi}{8}\\)
    \nAnswer:
    \n\"1st
    \nQuestion 67.
    \nProve that
    \n\\( \\cos ^{2} x+\\cos ^{2}\\left(x+\\frac{\\pi}{3}\\right)+\\cos ^{2}\\left(x-\\frac{\\pi}{3}\\right)=\\frac{3}{2}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 68.
    \n\\(\\begin{aligned} &\\text { If } \\tan x=\\frac{3}{4}, \\pi<x<\\frac{3 \\pi}{2}, \\text { find the value of }\\\\ &\\sin \\frac{x}{2}, \\cos \\frac{x}{2} \\text { and } \\tan \\frac{x}{2} \\end{aligned}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 69.
    \nFind \\(\\sin \\frac{x}{2}, \\cos \\frac{x}{2} \\text { and } \\tan \\frac{x}{2} \\)
    \n(i) \\(\\tan x=-\\frac{4}{3}, \\) x in quadrant II
    \n(ii) \\( \\cos x=-\\frac{1}{3}, \\) x in quadrant III
    \n(iii) \\( \\sin x=\\frac{1}{4}, x \\)in quadrant II
    \nAnswer:
    \n\"1st
    \n\"1st
    \n\"1st
    \n\"1st<\/p>\n

    Trigonometric Equations<\/strong><\/p>\n

    Question 1.
    \nDefine trigonometric equations.
    \nAnswer :
    \nEquations involving trigonometric functions of a variable are called trigonometric equations.<\/p>\n

    Question 2.
    \nDefine principal solutions of trigonometric equations.
    \nAnswer :
    \nThe solutions of a trigonometric equation for which 0 \u2264 x \u2264 2\u03c0 are called principal solutions.<\/p>\n

    \"KSEEB<\/p>\n

    Question 3.
    \nDefine general solution of trigonometric equation.
    \nAnswer :
    \nThe expression involving integer \u2018n\u2019 which gives all solutions of a trigonometric equation is called the general solution.<\/p>\n

    Question 4.
    \nFind the principal solution of the following equations
    \n(i) \\(\\sin x=\\frac{\\sqrt{3}}{2}\\)
    \n(ii) \\(\\tan x=-\\frac{1}{\\sqrt{3}}\\)
    \n(iii) \\( \\tan x=\\sqrt{3}\\)
    \n(iv) sec x=2
    \n(v) \\( \\cot x=\\sqrt{3}\\)
    \n(vi) cosec x=-2
    \nAnswer:
    \n\"1st
    \n\"1st
    \n\"1st<\/p>\n

    Question 5.
    \nProve that, for any real numbers x and y, sin x = sin y implies x =n\u03c0 + (-1)n<\/sup> y , where n\u2208Z
    \nAnswer:
    \n\"1st<\/p>\n

    Question 6.
    \nProve that, for any real numbers x and y, cosx=cosy, implies x = 2n\u03c0\u00b1y,where n\u2208Z
    \nAnswer:
    \n\"1st<\/p>\n

    Question 7.
    \nProve that, if x and y are not odd multiple of \\(\\frac{\\pi}{2} \\)
    \nAnswer:
    \nGiven: tan x=tan y
    \n\"1st<\/p>\n

    Question 8.
    \nFind the solution of\u00a0 \\(\\sin x=-\\frac{\\sqrt{3}}{2}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Note:
    \n\\(\\frac{4 \\pi}{3}\\) is one such value of x for which \\(\\sin x=-\\frac{\\sqrt{3}}{2} \\)
    \nOne may take any other value of x for which \\(\\sin x=-\\frac{\\sqrt{3}}{2}\\)
    \nThe solutions obtained will be the same although these may apparently look different.<\/p>\n

    Question 9.
    \nSolve: \\(\\cos x=\\frac{1}{2}\\)
    \nAnswer:
    \n\\(\\begin{array}{l}{\\text { We have, } \\cos x=\\frac{1}{2}=\\cos \\frac{\\pi}{3}} \\\\ {x=2 n \\pi \\pm \\frac{\\pi}{3}, n \\in Z}\\end{array}\\)<\/p>\n

    Question 10.
    \nSolve:
    \n\\(\\tan 2 x=-\\cot \\left(x+\\frac{\\pi}{3}\\right)\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 11.
    \nFind the general solution for which cos4x = cos2x.
    \nAnswer:
    \n\"1st<\/p>\n

    Question 12.
    \nSolve: sin 2x – sin4x + sin 6x = 0
    \nAnswer :
    \n\"1st<\/p>\n

    Question 13.
    \nSolve: cos3x + cosjc-cos2jc = 0.
    \nAnswer :
    \nGiven: (cos 3x + cosx) – cos 2x = 0
    \n\u21d2 2cos2x cosac-cos 2x = 0
    \ncos 2x(2cos x -1) = 0
    \n\"1st<\/p>\n

    Question 14.
    \nSolve: sin2x+cosx = 0.
    \nAnswer :
    \nGiven sin 2x + cosx = 0
    \n\u21d2 2sinxcosx + cosx = 0
    \n\u21d2 cosx(2sin x+1)=0
    \n\"1st<\/p>\n

    Question 15.
    \nSolve: sinx + sin3x+sin5x = 0.
    \nAnswer :
    \nGiven: (sin5x + sinx) + sin3x = 0
    \n\u21d22sin3xcos2x + sin3x = 0
    \n\u2234 sin3x(2cos2x + 1) = 0
    \n\"1st<\/p>\n

    Question 16.
    \nSolve: 2cos2<\/sup>x + 3sinx = 0
    \nAnswer :
    \nGiven: 2(1 – sin2<\/sup> x) + 3sin x = 0
    \n\u21d2 2sin2<\/sup> x-3sinx-2 = 0
    \n\u21d2 (2sin x+1)(sin x-2)=0
    \n\"1st<\/p>\n

    Question 17.
    \nSolve: sec2<\/sup>2x = 1-tan2x.
    \nAnswer :
    \n\"1st<\/p>\n

    Sine and cosine formulae<\/strong><\/p>\n

    Question 1.
    \nState and prove sine formula.
    \nAnswer :
    \nStatement: In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC
    \n\\(\\frac{\\sin A}{a}=\\frac{\\sin B}{b}=\\frac{\\sin C}{c}\\)
    \nProof : Let ABC be either acute angled triangle or obtuse angled triangle.
    \n\"1st
    \nThe altitude h is drawn from the vertex B to meet the side AC in point D.
    \nFrom the right angled triangle ABD, we have<\/p>\n

    \"1st<\/p>\n

    Question 2.
    \nState and prove cosine formula.
    \nAnswer :
    \nStatement: Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B and C respectively, then
    \na2<\/sup> =b2<\/sup> + c2<\/sup>-2bc cos A
    \nb2<\/sup> =c2<\/sup> +a2<\/sup>-2ca cos B
    \nc2<\/sup> =a2<\/sup> +b2<\/sup> – 2ab cos C
    \nProof:
    \nLet ABC be triangle as given below
    \n\"1st
    \nReferring to second triangle ,we have
    \nBC2<\/sup> = BD2<\/sup> + DC2<\/sup> = BD2<\/sup> + (AC – AD)2
    \n<\/sup>= BD2<\/sup> + AD2<\/sup> + AC2<\/sup> – 2AC AD
    \n= AB2<\/sup> + AC2<\/sup> -2AC AB cos A
    \n= a2<\/sup> =b2<\/sup> +c2<\/sup> -2bc cos A
    \nSimilarly we can obtain
    \nb2<\/sup> =c2<\/sup> + a2<\/sup> – 2ca cosB
    \nand c2<\/sup> = a2<\/sup> +b2<\/sup> – 2ab cos C<\/p>\n

    Keen eye:
    \n\\(\\cos A=\\frac{b^{2}+c^{2}-a^{2}}{2 b c}\\)
    \n\\(\\cos B=\\frac{c^{2}+a^{2}-b^{2}}{2 a c} \\)
    \n\\(\\cos C=\\frac{a^{2}+b^{2}-c^{2}}{2 a b}\\)<\/p>\n

    \"KSEEB<\/p>\n

    Question 3.
    \n(Napier Analogies) In triangle ABC, .prove that
    \n(i)\\(\\tan \\frac{B-C}{2}=\\frac{b-c}{b+c} \\cot \\frac{A}{2}\\)
    \n(ii) \\(\\tan \\frac{B-C}{2}=\\frac{b-c}{b+c} \\cot \\frac{A}{2}\\)
    \n(iii) \\(\\tan \\frac{B-C}{2}=\\frac{b-c}{b+c} \\cot \\frac{A}{2}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 4.
    \nIn any triangle ABC , if a = 18, b = 24, c = 30, find
    \n(i) cosA, cosB, cosC
    \n(ii) sinA, sinB, sinC
    \nAnswer :
    \n(i) We have
    \n\"1st<\/p>\n

    Question 5.
    \nFor any triangle ABC,Prove that
    \n\\(\\frac{a+b}{c}=\\frac{\\cos \\left(\\frac{A-B}{2}\\right)}{\\sin \\frac{C}{2}}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 6.
    \nFor any triangle ABC, Prove that
    \n\\(\\frac{a-b}{c}=\\frac{\\sin \\left(\\frac{A-B}{2}\\right)}{\\cos \\frac{C}{2}}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 7.
    \nFor any triangle ABC, Prove that
    \n\\(\\sin \\left(\\frac{B-C}{2}\\right)=\\frac{b-c}{a} \\cos \\frac{A}{2}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 8.
    \nIn any \u0394ABC, prove that a(bcosC-c cos B) = b2<\/sup>-c2<\/sup>.
    \nAnswer :
    \n\"1st<\/p>\n

    Question 9.
    \nIn any \u0394ABC, prove that
    \n\\( a(\\cos C-\\cos B)=2(b-c) \\cos ^{2} \\frac{A}{2}\\)
    \nAnswer:
    \n\"1st
    \n\"1st<\/p>\n

    Question 10.
    \nIn any \u0394ABC, prove that
    \n\\(\\frac{\\sin (B-C)}{\\sin (B+C)}=\\frac{b^{2}-c^{2}}{a^{2}} \\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 11.
    \nIn any \u0394ABC, prove that
    \n\\((b+c) \\cos \\frac{B+C}{2}=a \\cos \\frac{B-C}{2}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 12.
    \nIn any \u0394ABC, prove that
    \nacosA+bcosB + ccosC = 2asin BsinC.
    \nAnswer :
    \nLHS – k sin A cos A + k sin B cos B + k sin C cos C
    \n\"1st<\/p>\n

    Question 13.
    \nIn any \u0394ABC, prove that
    \n\\(\\frac{\\cos A}{a}+\\frac{\\cos B}{b}+\\frac{\\cos C}{c}=\\frac{a^{2}+b^{2}+c^{2}}{2 a b c}\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 14.
    \nIn any \u0394ABC, prove that
    \n\\( \\left(b^{2}-c^{2}\\right) \\cot A+\\left(c^{2}-a^{2}\\right) \\cot B+\\left(a^{2}-b^{2}\\right) \\cot C=0\\)
    \nAnswer:
    \n\"1st<\/p>\n

    \"KSEEB<\/p>\n

    Question 15.
    \nIn any \u0394ABC, prove that
    \n\\( \\frac{b^{2}-c^{2}}{a^{2}} \\sin 2 A+\\frac{c^{2}-a^{2}}{b^{2}} \\sin 2 B+\\frac{a^{2}-b^{2}}{c^{2}} \\sin 2 C=0\\)
    \nAnswer:
    \n\"1st<\/p>\n

    Question 16.
    \nIn any \u0394ABC, prove that
    \nasin(B -C) + bsin(C – A) + csin(A-B) = 0
    \nAnswer :
    \nLHS = k sin A sin(B -C) + ksinB sin(C – A) +k sin C sin( A – B)
    \n= k[sin(B + C) sin(B – C)+sin(C + A) sin(C – A) +sin( A + B) sin( A – B)]
    \n= K[sin2<\/sup> B – sin2<\/sup> C + sin2<\/sup> C -sin2<\/sup> A + sin2<\/sup> A – sin2<\/sup> B]
    \n= k x 0
    \n=0=RHS.<\/p>\n

    Question 17.
    \nA tree stands vertically on a hill side which makes an angle of 15\u00b0 with the horizontal. From a point on the ground 35 m down the hill from the base of the tree, the angle of elevation of the top of the tree is 60\u00b0. Find the height of the tree.
    \nAnswer :
    \nIn \u0394OBT,
    \n\\( \\frac{h}{\\sin 45^{\\circ}}=\\frac{35}{\\sin 30^{\\circ}}\\)
    \n\"1st<\/p>\n

    Question 18.
    \nTwo ships leave a port at the same time. One goes 24 km per hour in the direction N45\u00b0E and other travels 32 km per hour in the direction S75\u00b0E. Find the distance between the ships at the end of 3 hours.
    \nAnswer :
    \nLet C be the port A and B are the positions of the two ships after 3 hours.
    \n\"1st<\/p>\n

    Question 19.
    \nTwo trees, A and B are on the same side of a river. From a point C in the river the distance of the trees A and B is 250 m and 300 m, respectively. If the angle C is 45\u00b0, find the distance of the trees.
    \nAnswer :
    \nGiven AC = 250 m
    \nBC = 300 m
    \n\"1st<\/p>\n

    Question 20.
    \nThe angle of elevation of the top point P of the vertical tower PQ of height h from a point A is 45\u00b0 and from a point B, the angle of elevation is 60\u00b0, where B is a point at a distance \u2018d\u2019 from the point A measured along the line AB which makes an angle 30\u00b0 with AQ. Prove that \\(d=h(\\sqrt{3}-1) \\)
    \nAnswer:
    \n\"1st
    \n\"1st<\/p>\n

    Question 21.
    \nA lamp past is situated at the middle point M of the side AC of a triangular plot ABC with BC = 7 m, CA = 8 m and AB = 9 m. Lamp post subtends an angle 15\u00b0 at the point B. Determine the height of the lamp post.
    \nAnswer:
    \n\"1st
    \n\"1st<\/p>\n","protected":false},"excerpt":{"rendered":"

    Students can Download Maths Chapter 3 Trigonometric Functions Questions and Answers, Notes Pdf, 1st PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 1st PUC Maths Question Bank Chapter 3 Trigonometric Functions Measurement of Angles: Question 1. What is the …<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[81],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/12650"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=12650"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/12650\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=12650"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=12650"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=12650"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}