Trigonometric Ratios & Identities · Mathematics · AP EAPCET
MCQ (Single Correct Answer)
$$ \sin \frac{\pi}{12} \sin \frac{2 \pi}{12} \sin \frac{3 \pi}{12} \sin \frac{4 \pi}{12} \sin \frac{5 \pi}{12} \sin \frac{6 \pi}{12}= $$
If $A+B+C+D=2 \pi$, then $\sin A+\sin B+\sin C+\sin D=$
If $\cos x+\sin x=\frac{1}{2}$ and $0
If $\sin \theta+2 \cos \theta=1$ and $\theta$ belongs to 4 th quadrant (not lying on the coordinate axes), then $7 \cos \theta+6 \sin \theta=$
If $A$ and $B$ are acute angles satisfying $3 \cos ^2 A+2 \cos ^2 B=4$ and $\frac{3 \sin A}{\sin B}=\frac{2 \cos B}{\cos A}$, then $A+2 B=$
$$ \begin{aligned} & \left(4 \cos ^2 \frac{\pi}{20}-1\right)\left(4 \cos ^2 \frac{3 \pi}{20}-1\right) \\ & \left(4 \cos ^2 \frac{5 \pi}{20}+1\right)\left(4 \cos ^2 \frac{7 \pi}{20}-1\right)\left(4 \cos ^2 \frac{9 \pi}{20}-1\right)= \end{aligned} $$
If $A$ and $B$ are the values such that $(A+B)$ and $(A-B)$ are not odd multiples of $\frac{\pi}{2}$ and $2 \tan (A+B)=3 \tan (A-B)$, then $\sin A \cos A=$
If $\cos ^3 80^{\circ}+\cos ^3 40^{\circ}-\cos ^3 20^{\circ}=k$, then $\frac{4 k}{3}=$
$$ \cos 13^{\circ} \sin 17^{\circ} \sin 21^{\circ} \cos 47^{\circ}= $$
$\operatorname{cosec} 48^{\circ}+\operatorname{cosec} 96^{\circ}+\operatorname{cosec} 192^{\circ}+\operatorname{cosec} 384^{\circ}=$
If $\cos \theta=\frac{-3}{5}$ and $\theta$ does not lie in second quadrant, then $\tan \frac{\theta}{2}=$
If $\alpha$ is the maximum value and $\beta$ is the minimum value of $\cos ^2 \frac{x}{4}+\sin \frac{x}{4}, x \in R$, then $\alpha-\beta=$
If $A$ and $B$ are positive acute angles satisfying $3 \cos ^2 A+2 \cos ^2 B=4$ and $\frac{3 \sin A}{\sin B}=\frac{2 \cos B}{\cos A}$, then $A+2 B=$
If $\sin x-\sin y=\frac{27}{65}$ and $\cos x-\cos y=\frac{-21}{65}$, then $\sin (x+y)=$
If $\left(\frac{\sin 3 \theta}{\sin \theta}\right)^2-\left(\frac{\cos 3 \theta}{\cos \theta}\right)^2=a \cos b \theta$, then $a: b=$
An aeroplane is flying at a constant speed, parallel to the horizontal ground at a height of 5 kms . A person on the ground observed that the angle of elevation of the plane is changed from $15^{\circ}$ to $30^{\circ}$ in the duration of 50 seconds, then the speed of the plane (in kmph ) is
If $A+B=\frac{\pi}{4}$, then $\frac{\cos B-\sin B}{\cos B+\sin B}=$
If $7 \cos \theta-\sin \theta=5$ and $\tan \theta>0$, then $\tan \theta=$
$$ \sin ^3 10^{\circ}+\sin ^3 50^{\circ}-\sin ^3 70^{\circ}= $$
$$ \begin{aligned} \frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\frac{1}{\sin 3^{\circ} \sin 4^{\circ}} & +\frac{1}{\sin 89^{\circ} \sin 90^{\circ}}= \end{aligned} $$
$$ \cos ^3 \frac{\pi}{8} \cos \frac{3 \pi}{8}+\sin ^3 \frac{\pi}{8} \sin \frac{3 \pi}{8}= $$
If $A+B+C=\frac{\pi}{4}$, then $\sin 4 A+\sin 4 B+\sin 4 C=$
If $630^{\circ}<\theta<810^{\circ}$ and $\tan \theta=-\frac{7}{24}$, then $\cos \left(\frac{\theta}{4}\right)=$
For $\theta \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ if $2 \cos \theta+\sin \theta=1$ and $7 \cos \theta+6 \sin \theta=k$, then the possible values of $k$ are
$$ \sum\limits_{k=0}^{12} \frac{1}{\sin \left((k+1) \frac{\pi}{6}+\frac{\pi}{4}\right) \sin \left(\frac{k \pi}{6}+\frac{\pi}{4}\right)}= $$
If $\cos \alpha=\sec h \beta$, then $\beta=$
$$ \tan ^2 \frac{\pi}{16}+\tan ^2 \frac{2 \pi}{16}+\tan ^2 \frac{3 \pi}{16}+\tan ^2 \frac{4 \pi}{16} $$
$+\tan ^2 \frac{5 \pi}{16}+\tan ^2 \frac{6 \pi}{16}+\tan ^2 \frac{7 \pi}{16}$ is equal to
$$ \begin{aligned} & \sin ^2 18^{\circ}+\sin ^2 24^{\circ}+\sin ^2 36^{\circ}+\sin ^2 42^{\circ}+\sin ^2 78^{\circ} \\ & +\sin ^2 90^{\circ}+\sin ^2 96^{\circ}+\sin ^2 102^{\circ}+\sin ^2 138^{\circ}+\sin ^2 162^{\circ} \text { is } \\ & \text { equal to } \end{aligned} $$
If $\sinh x=\frac{\sqrt{21}}{2}$, then $\cosh 2 x+\sinh 2 x$ is equal to
If $M_1$ and $M_2$ are the maximum values of $\frac{1}{11 \cos 2 x+60 \sin 2 x+69}$ and $3 \cos ^2 5 x+4 \sin ^2 5 x$ respectively, then $\frac{M_1}{M_2}=$
$$ 4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2 \pi}{7} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{7}= $$
Assertion (A) : If $A=10^{\circ}, B=16^{\circ}$ and $C=19^{\circ}$, then $\tan 2 A \tan 2 B+\tan 2 B \tan 2 C+\tan 2 C \tan 2 A=1$
Reason (R) : If $A+B+C=180^{\circ}, \cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}$
$$ =\cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2} $$
Which of the following is correct ?
Statement $(\mathrm{S} 1) \sin 55^{\circ}+\sin 53^{\circ}-\sin 19^{\circ}-\sin 17^{\circ}=\cos 2^{\circ}$
Statement (S2) Range of $\frac{1}{3-\cos 2 x}$ is $\left[\frac{1}{4}, \frac{1}{2}\right]$
Which one of the following is correct?
If $$\sin ^4 \theta \cos ^2 \theta=\sum_\limits{n=0}^{\infty} a_{2 n} \cos 2 n \theta$$, then the least $$n$$ for which $$a_{2 n}=0$$ is
If $$\sin \theta=-\frac{3}{4}$$, then $$\sin 2 \theta=$$
$$\begin{aligned} & \frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\ldots +\frac{1}{\sin 89^{\circ}+\sin 90^{\circ}}= \end{aligned}$$
Which of the following trigonometric values are negative?
I. $$\sin \left(-292^{\circ}\right)$$
II. $$\tan \left(-190^{\circ}\right)$$
III. $$\cos \left(-207^{\circ}\right)$$
IV. $$\cot \left(-222^{\circ}\right)$$
$$\sin ^2 \frac{2 \pi}{3}+\cos ^2 \frac{5 \pi}{6}-\tan ^2 \frac{3 \pi}{4}=$$
A true statement among the following identities is
If $$A+B+C=\pi, \cos B=\cos A \cos C$$, then $$\tan A \tan C=$$
The value of $$\tan \left(\frac{7 \pi}{8}\right)$$ is
$$1+\sec ^2 x \sin ^2 x=$$
If the identity $$\cos ^4 \theta=a \cos 4 \theta+b \cos 2 \theta+c$$ holds for some $$a, b, c \in Q$$ then $$(a, b, c)=$$
The value of $$\frac{\sin \theta+\sin 3 \theta}{\cos \theta+\cos 3 \theta}$$ is
If $$(1+\tan 1^{\circ})(1+\tan 2^{\circ}) \ldots(1+\tan 45^{\circ})=2^n,$$ then $$n=$$
$$\frac{\cos \theta}{1-\tan \theta}+\frac{\sin \theta}{1-\cot \theta}=$$
If $$\operatorname{cosech} x=\frac{4}{5}$$, then $$\sinh x=$$
Let $$\theta$$ be an angle in the standard position such that the point $$(-5,12)$$ lies on its terminal side, then
If $$\cos \frac{\pi}{4} \cos \frac{\pi}{8} \cos \frac{\pi}{16} \cos \frac{\pi}{32}=2^m \operatorname{cosec} \frac{\pi}{n}$$, then $$m+n$$ is equal to
If $$A+B+C=\frac{3 \pi}{2}$$, then $$\cos 2 A+\cos 2 B+\cos 2 C$$ is equal to
$$\sinh (x+y) \cosh (x-y)$$ is equal to
What is the value of $$\cos \left(22 \frac{1}{2}\right)^{\circ}$$ ?
If $$\cos \theta=-\sqrt{\frac{3}{2}}$$ and $$\sin \alpha=\frac{-3}{5}$$, where '$$\theta$$' does not lie in the third quadrant, then the value of $$\frac{2 \tan \alpha+\sqrt{3} \tan \theta}{\cot ^2 \theta+\cos \alpha}$$ is equal to
If $$\tan \beta=\frac{\tan \alpha+\tan \gamma}{1+\tan \alpha \tan \gamma}$$, then $$\frac{\sin 2 \alpha+\sin 2 \gamma}{1+\sin 2 \alpha \sin 2 \gamma}$$ is equal to
The sides of a triangle inscribed in a given circle subtend angles $$\alpha, \beta, \gamma$$ at the center. The minimum value of the AM of $$\cos \left(\alpha+\frac{\pi}{2}\right), \cos \left(\beta+\frac{\pi}{2}\right)$$ and $$\cos \left(\gamma+\frac{\pi}{2}\right)$$ is equal to
In a $$\triangle A B C$$, if $$3 \sin A+4 \cos B=6$$ and $$4 \sin B+3 \cos A=1$$, then $$\sin (A+B)$$ is equal to
$$\tan \alpha+2 \tan 2 \alpha+4 \tan 4 \alpha+8 \cot 8 \alpha$$ is equal to
If $$f(x)=\frac{\cot x}{1+\cot x}$$ and $$\alpha+\beta=\frac{5 \pi}{4}$$, then the value of $$f(\alpha) f(\beta)$$ is equal to
In $$\triangle A B C \cdot \frac{a+b+c}{B C+A B}+\frac{a+b+c}{A C+A B}=3$$, then $$\tan \frac{C}{8}$$ is equal to
Mean of the values $$\sin ^2 10 Y, \sin ^2 20 Y, \sin ^2 30 Y, \ldots \ldots \ldots ., \sin ^2 90 Y$$ is
When the coordinate axes are rotated through an angle 135$$\Upsilon$$, the coordinates of a point $$P$$ in the new system are known to be $$(4,-3)$$. Then find the coordinates of $$P$$ in the original system.
The maximum value of $$f(x)=\sin (x)$$ in the interval $$\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$$ is
$$\tan 2 \alpha \cdot \tan (30 Y-\alpha)+\tan 2 \alpha \cdot \tan (60 Y-\alpha)+\tan (60 \Upsilon-\alpha) \cdot \tan (30 \gamma-\alpha)$$ is equal to
If $$\sin \alpha - \cos \alpha = m$$ and $$\sin 2\alpha = n - {m^2}$$, where $$ - \sqrt 2 \le m \le \sqrt 2 $$, then n is equal to
If $$\sinh u=\tan \theta$$, then $$\cosh u$$ is equal to