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Trigonometric Functions questions





Question 1.
Find the value of $\sqrt 3 cosec 20^0- sec 20^0$

Question 2.
Find the value of $(1 + \cos \frac {\pi}{8})(1 + \cos \frac {3\pi}{8})(1 + \cos \frac {5\pi}{8})(1 + \cos \frac {7\pi}{8})$

Question 3.
If $a \cos \theta + b \sin \theta = m$ and $a \sin \theta - b \cos \theta = n$, then show that $a^2+ b^2= m^2+ n^2$

Question 4.
if $tan (\alpha)= \frac {m}{m+1}$ and $tan (\beta)= \frac {1}{2m+1}$, Prove that $\alpha + \beta = \frac {\pi}{4}$

Question 5.
Prove that $5 cos \theta + 3 cos(\theta + \frac {\pi}{3}) + 3$ lies between -4 and 10

Question 6.
if $ \frac {\pi}{2} \leq x \leq \pi $ and tan x =-4/3 , Find the value of sin (x/2),cos (x/2),tan (x/2)

Question 7.
Prove that a. $\sin 12^0 \sin 48^0 \sin 54^0= \frac {1}{8}$
b. $\sin 20^0 \sin 40^0 \sin 60^0 \sin 80^0= \frac {3}{16}$

Question 8.
If $a \cos (2 \theta) + b \sin (2\theta) = c$ has $\alpha$ and $\beta$ as its roots, then prove that
$\tan (\alpha) + \tan (\beta) =\frac {2b}{a+c}$

Question 9.
Prove that $cot 7 \frac {1}{2} ^0= \sqrt 2 + \sqrt 3 + \sqrt 4 + \sqrt 6$

Question 10.
if $\alpha$ and $\beta$ are the solutions of $sin^2 x + a sin x+ b=0$ as well as that of $cos^2 x + c cos (x) +d=0$,then prove that
$sin (\alpha + \beta) = \frac {2ac}{a^2 + c^2}$

Question 11.
If $\alpha$ and $\beta$ are the solutions of the equation $a tan \theta + b sec \theta = c$,
then show that $tan (\alpha + \beta) = \frac {2ac}{a^2 - c^2}$

Question 12.
Show that
$\frac {sec^2 \theta - tan \theta}{sec^2 \theta + tan \theta}$ lies between 1/3 and 3

Question 13.
In any $\Delta ABC$ , if $a^2, b^2 ,c^2$ are in A.P,Prove that cot A,cot B and Cot C are in A.P

Question 14.
Prove that
$2 cos(x) - cos(3x) -cos (5x) = 16 cos^3 (x) sin^2 (x)$

Question 15.
Determine the period and Graph of $y=sin (2x)$

Question 16.
Determine the period and Graph of $y=-2 cos(x) + 3$

Question 17.
Determine the period and Graph of $y=\frac {1}{2} cos (x)$

Question 18.
Sketch the Graph of $y=3 sin (x)$

Question 19.
Prove that
$\frac {cos (7x) + cos (5x)}{sin (7x) -sin (5x)}=cot (x)$

Question 20.
Find the value of $\tan (\frac {13 \pi}{10})$


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