### Mathematics Assignments | PDF | 8 to 12

PDF Files of Mathematics Assignments From VIII Standard to XII Standard PDF of mathematics Assignments for the students from VIII standard to XII standard.These assignments are strictly according to the CBSE and DAV Board Final question Papers

## Trigonometric Functions

Extra questions of chapter 3 class 11 Trigonometric Functions with answer and  hints to the difficult questions. Important and useful math. assignment for the students of class 11

For better results

• Students should learn all the basic points of Trigonometry up to 11th standard
• Student should revise NCERT book thoroughly with examples.
• Now revise this assignment. This assignment integrate the knowledge of the students.

ASSIGNMENT FOR XI  STANDARD TRIGONOMETRY

STRICTLY ACCORDING TO THE CBSE AND DAV BOARD

Question 1

Find the degree measure for the following radian measure

$i)\:\; \; \frac{2\pi }{15}Radian\; \; \; \; \; ............. \; \; [Ans\; \; 24^{o}]$

$ii)\:\; \; \frac{11 }{16}Radian\; \; \; \; \; ............. \; \; [Ans\; \; 39^{o}\: 22'\: 30'']$

$iii)\:\; \; (-2)Radian\; \; \; \; \; ............. \; \; [Ans\; \; -(114^{o}\: 32'\: 43'')]$

Question 2

Find the radian measure for the following degree measure

$i)\:\; \; -37^{o}30'\; \; \; \; \; ............. \; \; [Ans\; \; -\left ( \frac{5\pi }{24} \right )]$

$ii)\:\; \; 40^{o}\: 20'\; \; \; \; \; ............. \; \; [Ans\; \; \left ( \frac{121\pi }{540} \right )]$

$iii)\:\; \; 5^{o}\: 37'\:30'' \; \; \; \; \; ............. \; \; [Ans\; \; \left ( \frac{\pi }{32} \right )]$

Question 3

Find the magnitude, radian and degree, of the interior angles of a regular

$i)\:\; \; Pentagon \; \; \; \; \; ............. \; \; [Ans\; \; \left ( \frac{3\pi }{5}\; :\; 108^{o} \right )]$

$ii)\:\; \; Octagon \; \; \; \; \; ............. \; \; [Ans\; \; \left ( \frac{3\pi }{4}\; :\; 135^{o} \right )]$

$iii)\; \; \; Heptagon \; \; \; \; \; ............. \; \; Ans\; \; \left ( \frac{5\pi }{7}\; :\; 128^{o}\; 34'\; 17'' \right )$

Solution Hint:

Note:  Each interior angle of a regular polygon is given by :

$\frac{(n-2)}{n}\times 180$ Where n is the number of sides.

Question 4

If sin Î¸ =12/13 and Î¸ lie in the second quadrant, then find the value of  sec Î¸ + tan Î¸.     Ans. [- 5]

Question 5

Prove the followings

i)   cos24o + cos 55o + cos 125o + cos 204o + cos 300o = 1/2

ii)  sin 600o tan(-690o) + sec 840o cot(-945o) = 3/2

iii) $sin^{2}\left \{ \frac{\pi }{8} \right \} +sin^{2}\left \{ \frac{3\pi }{8} \right \}+sin^{2}\left \{ \frac{5\pi }{8} \right \}+sin^{2}\left \{ \frac{7\pi }{8} \right \}=2$

Question 6:   Simplify

$tan\left ( \frac{\pi }{20} \right )tan\left ( \frac{3\pi }{20} \right )tan\left ( \frac{5\pi }{20} \right )tan\left ( \frac{7\pi }{20} \right )tan\left ( \frac{9\pi }{20} \right )$

Ans: 1

Solution Hint:

$tan\left ( \frac{\pi }{20} \right )tan\left ( \frac{3\pi }{20} \right )tan\left ( \frac{\pi }{4} \right )tan\left ( \frac{10\pi-3\pi }{20} \right )tan\left ( \frac{10\pi-\pi }{20} \right )$

$=tan\left ( \frac{\pi }{20} \right )tan\left ( \frac{3\pi }{20} \right ).1.tan\left ( \frac{\pi}{2}-\frac{3\pi}{20} \right )tan\left ( \frac{\pi}{2}-\frac{\pi}{20} \right )$

$=tan\left ( \frac{\pi }{20} \right )tan\left ( \frac{3\pi }{20} \right )cot\left ( \frac{3\pi}{20} \right )cot\left (\frac{\pi}{20} \right ) = 1$

Question 7

Evaluate the following :

$i) \; \; tan105^{o}\; \; ......\; \; Ans \left [ \frac{\sqrt{3}+1}{1-\sqrt{3}} \right ]$

$ii) \; \; cos105^{o}+sin105^{o}\; \; ......\; \; Ans \left [ \frac{1}{\sqrt{2}} \right ]$

Question 8

Prove that : tan70o = tan20o + 2tan50o

Solution Hint:

$70^{o}=50^{o}+20^{o}\;$

$tan\: 70^{o}=tan(50^{o}+20^{o})$

$tan\: 70^{o}=\frac{tan50^{o}+tan20^{o}}{1-tan50^{o}\; tan20^{o}}$

Now cross-multiply and simplify the above fraction we get the required result.

Question 9

$Prove \; that: \frac{cos9^{o}+sin9^{o}}{cos9^{o}-sin9^{o}}=tan54^{o}$

Question 10

If tan A = k tan B, then show  that: $sin(A+B)=\left (\frac{k+1}{k-1} \right )sin(A-B)$

Solution Hint: tan A = k tan B

$\frac{sinA}{cosA}=k\; \frac{sinB}{cosB}$

$\Rightarrow \frac{sinA\: cosB}{cosA\: sinB}=\frac{k}{1}$

Now apply componendo and dividendo and using trigonometric formulas.

Question 11

If tan(A + B) = p, tan(A – B) = q, then show that :  $tan\: 2A = \frac{p+q}{1-pq}$

Solution Hint:

Start from RHS and then putting the value of p and q then simplify and get the result.

Question 12

An angle Î± is divided into two parts such that the ratio of the tangents of the two parts = k  and difference of two parts = x  then show that:

$sin\: x=\frac{k-1}{k+1}\: sin\alpha$

Solution hint:

Let two parts of  Î± are  p and q, Then

ATQ :  p + q = Î±,  p – q = x,  $\frac{tanp}{tanq}=\frac{k}{1}$

Now applying componendo and dividendo and simplify we get the required result.

Question 13

Prove that :   cos20o cos40o cos60o cos80o = 1/16

Question 14

Prove that :   sin20o sin40o sin60o sin80o = 3/16

Question 15

Prove That : $tan20^{o}\; tan40^{o}\; tan80^{o}=tan60^{o}$

Solution Hint :

Use tan Î¸ = sin Î¸/cos Î¸, then using AB and CD formulas

Question 16

If cos(Î¸ + 2 Î±) = n cos Î¸ , then prove that : $cot\; \alpha =\frac{1+n}{1-n}\; tan(\theta +\alpha )$

Solution Hint: $\frac{cos\: \theta }{cos(\theta +2\alpha) }=\frac{1}{n}$

Applying componendo and dividendo  then applying CD formulas then simplify the fractions we get the required result.

Question 17

Prove that:   $sec\left ( \frac{\pi }{4}+\theta \right )sec\left ( \frac{\pi }{4}-\theta \right )=2\: sec\: 2\theta$

Solution Hint

Taking LHS and convert these into cosine functions.

Multiply and divide by 2 and the apply AB formulas

Question 18

Prove that:  $\frac{sin11AsinA+sin7Asin3A}{cos11AsinA+cos7Asin3A}=tan8A$

Solution Hint:

Multiply and divide by 2 and then apply AB formulas

Question 19

Find the value of sin18o

Ans: $\left [sin18^{o}=\frac{\sqrt{5}-1}{4} \right ]$

Solution:

Î¸ = 18o   5 Î¸ = 90o     2 Î¸ + 3 Î¸ = 90o    2 Î¸ = 90o - 3 Î¸

Now taking sin on both side we get

Sin (2 Î¸) = Sin (90o - 3 Î¸)    Sin (2 Î¸) = Cos (3 Î¸)

2Sin Î¸ cos Î¸ = 4Cos3 Î¸ – 3cos Î¸     2Sin Î¸  = 4Cos2 Î¸ – 3

2Sin Î¸  = 4(1-sin2 Î¸) – 3

4 sin2 Î¸ + 2sin Î¸ – 1 = 0

Now using quadratic formula here and find the value of sin Î¸

Question 20

Prove that :  $\frac{sec8A-1}{sec4A-1}=\frac{tan8A}{tan2A}$

Solution Hint:

$\frac{sec8A-1}{sec4A-1}=\frac{1-cos8A}{cos8A}\times \frac{cos4A}{1-cos4A}$

$=\frac{2sin^{2}4A.\: cos4A}{cos8A.\: 2sin^{2}2A}=\frac{(2sin4A\: cos4A)sin4A}{cos8A.\: 2sin^{2}2A}\$

$=\left (\frac{sin8A}{cos8A} \right ).\frac{2sin2Acos2A}{2sin^{2}A}=\frac{tan8A}{tan4A}$

Question 21

i) Evaluate:  $sin\left ( 22\frac{1}{2} \right )^{o}\: \: ...\: \: Ans\left [ \frac{\sqrt{2-\sqrt{2}}}{2} \right ]$

Solution Hint:  Using: $Sin\theta =\sqrt{\frac{1-cos2\theta }{2}},\; \; where\; \theta =\frac{45}{2}$

ii)  Evaluate:  $tan\left ( 22\frac{1}{2} \right )^{o}\: \: ...\: \: Ans\left [ \sqrt{2}-1 \right ]$

Question 22

Prove that:  $\frac{cos8Acos5A-cos12Acos9A}{sin8Acos5A+cos12Asin9A}=tan4A$

Solution Hint:

Multiply numerator and denominator by 2

Applying AB formulas we get

$\frac{cos13A-cos21A}{sin13A+sin21A}$

Now applying CD formulas we get the required result.

Question 23

Find  $sin\left ( \frac{x}{2} \right )$ , $cos\left ( \frac{x}{2} \right )$  and  $tan\left ( \frac{x}{2} \right )$ , when tan x = $-\frac{4}{3}$ , and x lie in II quadrant

Ans:

$sin\left ( \frac{x}{2} \right )=\frac{2}{\sqrt{5}}$

$cos\left ( \frac{x}{2} \right )=\frac{1}{\sqrt{5}}$  ,  $tan\left ( \frac{x}{2} \right )=2$

Question 24

Prove that:   $4sinxsin\left ( x+\frac{\pi }{3} \right )sin\left ( x+\frac{2\pi }{3} \right )=sin3x$
Solution Hint: Apply AB formulas

Question 25

If tan35o = Î± , then find the value of   $\frac{tan145^{o}-tan125^{o}}{1+tan145^{o}tan125^{o}}$   in terms of Î±

Ans:  $\frac{1-\alpha ^{2}}{2\alpha }$

Question 26

Prove that : $\frac{sin5x-2sin3x+sinx}{cos5x-cosx}=tanx$

Question: 27

Prove that : $sin^{2}x+sin^{2}\left ( x+\frac{\pi }{3} \right )+sin^{2}\left ( x-\frac{\pi }{3} \right )=\frac{3}{2}$

Solution Hint:

Use formula: $sin^{2}x=\frac{1-cos2x}{2}$

Now simplify and then apply CD formula we get the required result

Question 28

Prove that:  $tan4x=\frac{4tanx(1-tan^{2}x)}{1-6tan^{2}x+tan^{4}x}$

Question 29

If tan x + tan y + tan x tan y = 1, find (x + y).

Ans: x + y = 45o

Solution Hint:

tan x + tan y  = 1-tan x tan y

Dividing on both side by 1-tanx tany we get

$\frac{tanx+tany}{1-tanxtany}=1=tan45^{o}$

tan(x + y) = tan 45o  ⇒ x + y = 45o

Question: 30

Prove that: Cos6Î¸ = 32cos6 Î¸ - 48cos4 Î¸ + 18cos2 Î¸ - 1

Solution Hint: Use cos6Î¸ = cos3(2Î¸) = - 3cos2Î¸ + 4cos3 2Î¸ and then proceed.

Question: 31

Prove that: cos6x = 1 - 18sin2x + 48sin4x - 32sin6x

Solution Hint: Use cos6Î¸ = cos3(2Î¸) =  -3cos2Î¸ + 4cos3 2Î¸ .

Now putting cos2Î¸ = 1 - 2sin2Î¸ we get

cos6Î¸ = -3(1 - 2sin2Î¸) + 4(1 - 2sin2Î¸)3

Solving this we get the required result

Question 32

Prove that: $\frac{tan5\theta +tan3\theta }{tan5\theta -tan3\theta}=4cos2\theta cos4\theta$

Solution Hint

Using tanÎ¸ = sinÎ¸/cosÎ¸ in numerator and in denominator

Taking LCM then using the formula sin(A+B) and sin (A-B) in numerator and in denominator respectively.

Now using sin2Î¸ = 2sinÎ¸cosÎ¸ in the numerator for two times.

Question: 33

Prove that:  sin3x cos3x + cos3xsin3x =  $\frac{3}{4}$sin4x

Solution Hint: Expand sin3x and cos3x

Now multiply and divide by 2

Applying 2sinxcosx = sin2x for two times we get the required result.

Question 34

Prove that :  $cos\theta\; cos\frac{\theta }{2}-cos3\theta\; cos\frac{9\theta }{2} = sin4\theta \; sin\frac{7\theta}{2}$

Solution : $cos\theta.\; cos\frac{\theta }{2}-cos3\theta.\; cos\frac{9\theta }{2}$

$=\frac{1}{2}\left [2cos\theta.\; cos\frac{\theta }{2}-2cos3\theta.\; cos\frac{9\theta }{2} \right ]$

$=\frac{1}{2}\left [ cos\left ( \theta +\frac{\theta }{2} \right )+cos\left ( \theta -\frac{\theta }{2} \right ) \right ]- \frac{1}{2}\left [ cos\left ( 3\theta +\frac{9\theta }{2} \right )+cos\left ( 3\theta -\frac{9\theta }{2} \right ) \right ]\$

$=\frac{1}{2}\left [ cos\left (\frac{3\theta }{2} \right )+cos\left (\frac{\theta }{2} \right )-cos\left (\frac{15\theta }{2} \right )-cos\left (\frac{3\theta }{2} \right ) \right ]\$

$=\frac{1}{2}\left [ cos\left (\frac{\theta }{2} \right )-cos\left (\frac{15\theta }{2} \right ) \right ]$

$=\frac{1}{2}\left [ -2sin\frac{1}{2}\left ( \frac{\theta +15\theta }{2} \right ).\; sin\frac{1}{2}\left ( \frac{\theta -15\theta }{2} \right ) \right ]$

$=\frac{1}{2}\left [ -2sin4\theta .\; sin\left ( \frac{-7\theta }{2} \right ) \right ]$

$=sin4\theta .\; sin\left ( \frac{7\theta }{2} \right )$

Question 35

Evaluate: $If\; cos(\alpha +\beta )=\frac{4}{5}\; and\; sin(\alpha -\beta )=\frac{5}{13}$ and

$0<\alpha <\frac{\pi }{4}, \; \; Find \; \; tan2\alpha$

Solution:

$cos(\alpha +\beta )=\frac{4}{5}\Rightarrow sin(\alpha +\beta )=\frac{3}{5}$

$sin(\alpha -\beta )=\frac{5}{13}\Rightarrow cos(\alpha -\beta )=\frac{12}{13}$

$cos[(\alpha +\beta )+(\alpha -\beta )]=cos(\alpha +\beta)cos(\alpha -\beta)-sin(\alpha +\beta)sin(\alpha -\beta)$

$cos(2\alpha )=\frac{4}{5}\times \frac{12}{13}-\frac{3}{5}\times \frac{5}{13}=\frac{33}{65}$

$sin[(\alpha +\beta )+(\alpha -\beta )]=sin(\alpha +\beta)cos(\alpha -\beta)+cos(\alpha +\beta)sin(\alpha -\beta)$

$sin(2\alpha )=\frac{3}{5}\times \frac{12}{13}+\frac{4}{5}\times \frac{5}{13}=\frac{56}{65}$

$tan(2\alpha )=\frac{sin(2\alpha )}{cos(2\alpha )}=\frac{56/65}{33/65}=\frac{56}{33}$

Question: 36

Prove that: sin2Î± + sin2(Î± - Î²) - 2sinÎ± cosÎ² sin(Î± - Î²) = sin2Î²

Solution Hint:

LHS = sin2Î± + sin2(Î± - Î²) - [sin(Î± + Î²) + sin(Î± - Î²)]sin(Î± - Î²)

= sin2Î± + sin2(Î± - Î²) - sin(Î± + Î²)sin(Î± - Î²) - sin2(Î± - Î²)

= sin2Î± - [sin2Î± - sin2Î²] ..... [Using sin(A+B)sin(A-B) = sin2A - sin2B ]

= sin2Î²

Question 37

If  $xcos\theta =ycos\left ( \theta +\frac{2\pi }{3} \right )=zcos\left ( \theta +\frac{4\pi }{3} \right )$ , then find the value of xy + yz + zx

Solution Hint:

Let all equations = k

Now find the value of x, y, z in terms of k

Now find the value of  $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$  , it should be '0'

Putting this value in xy + yz + zx = $\frac{xyz}{z}+\frac{xyz}{x}+\frac{xyz}{y}$

=  $xyz\left [ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} \right ]$

= xyz ×0 = 0

Question 38

Find the value of : $\frac{tan^{2}15^{o}-1}{tan^{2}15^{o}+1}$

Ans: $-\frac{\sqrt{3}}{2}$

Solution Hint:

Taking "-" common from numerator

Using the formula:  $cos2\theta =\frac{1-tan^{2}\theta }{1+tan^{2}\theta }$

Question 39 : Prove that

$sin^{3}x+sin^{3}\left(\frac{2\pi}{3}+x\right)+sin^{3}\left(\frac{4\pi}{3}+x\right)=-\frac{3}{4}sin3x$

Solution Hint:

Using this formula: sin3x = 3sinx – 4sin3x

$\Rightarrow sin^{3}x=\frac{1}{4}[3sinx-sin3x]$

$LHS=\frac{3}{4}\left[sinx+sin\left(\frac{2\pi}{3}+x\right)+sin\left(\frac{4\pi}{3}+x\right)-sin3x\right]$

Using C, D formulas for the middle two terms we get

$LHS=\frac{3}{4}\left[sinx+2sin(\pi+x)cos\frac{\pi}{3}-sin3x\right]$

$LHS=\frac{3}{4}\left[sinx-2sin(x)\frac{1}{2}-sin3x\right]$
$LHS=\frac{3}{4}\left[sinx-sinx-sin3x\right]$
$LHS=-\frac{3}{4}sin3x=RHS$

General Solutions of Trigonometric Functions
These questions are deleted from the CBSE syllabus

Question 39

Find the general solution of the equation: cos x + cos 2x + cos 3x = 0.

Question 40

Solve: $cos\theta + cos2\theta + cos3\theta =0$

$Ans\left [ \theta =(2n+1)\frac{\pi }{4}, \; \; or \; \; 2n\pi \pm \frac{2\pi }{3} \right ]$

Question 41

Solve the equation: sin 3x + sin 5x + sin 7x = 0,   $\frac{\pi }{2}

Question 42

Solve: $cos3\theta +8cos^{3}\theta =0$

$Ans\left [ \theta =(2n+1)\frac{\pi }{2} \; or\; \; n\pi \pm \frac{\pi }{3}\right ]$

Question 43

Solve: $\sqrt{2}sec\; \theta +tan\; \theta =1$

Solution  Hint: $\sqrt{2}sec\; \theta +tan\; \theta =1$

$\sqrt{2}\; \frac{1}{cos\theta } +\frac{sin\theta }{cos\theta } =1\Rightarrow \sqrt{2}+sin\theta =cos\theta$

$cos\theta -sin\theta =\sqrt{2}$

$\frac{1}{\sqrt{2}}cos\theta-\frac{1}{\sqrt{2}}sin\theta=1$

$cos\frac{\pi }{4}\; cos\theta-sin\frac{\pi }{4}\; sin\theta=1$

$cos\left ( \theta +\frac{\pi }{4} \right )=1=cos\: 0$

$\theta +\frac{\pi }{4}=2n\pi \pm 0\; \Rightarrow \; \theta =2n\pi -\frac{\pi }{4}$

Question 44

Solve: $\sqrt{3}\; cosx-sinx=1$

$Ans\left [ x=2n\pi +\frac{\pi }{6}\; or\; 2n\pi -\frac{\pi }{2} \right ]$

Solution Hint:

$\sqrt{3}\; cosx-sinx=1$

Dividing on both side by

$\sqrt{(\sqrt{3})^{2}+(-1)^{2}}$ i.e. by 2

$\frac{\sqrt{3}}{2}\; cosx-\frac{1}{2}\; sinx=\frac{1}{2}$
$cos\frac{\pi }{6}\; cosx-sin\frac{\pi }{6}\; sinx=\frac{1}{2}$
Now proceed this question as previous one.

Question 45

Find the general solution of: 2 cos2x + 3 sin x = 0.

Question 46

Find general solution of cos2x cosec x + 3sinx + 3 = 0

Ans: $x=n\pi +(-1)^{n}\frac{3\pi }{2}$ ,   $x=n\pi +(-1)^{n}\frac{7\pi }{2}$

Solution Hint:

Convert all terms into sin x , we get a quadratic equation.

Solve the quadratic equations.

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THANKS FOR YOUR VISIT