Sample QuestionsTrigonometric Functions questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The value of $\frac{1-\tan^215^\circ}{1+\tan^215^\circ}$ is:
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$1$
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$\sqrt{3}$
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$\frac{\sqrt{3}}{2}$
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$2$
The value of $\sin\frac{\pi}{10}\sin\frac{13\pi}{10}$ is:
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$\frac{1}{2}$
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$-\frac{1}{2}$
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$-\frac{1}{4}$
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$1$
[Hint: Use $\sin18^\circ=\frac{\sqrt{5}-1}{4}$ and $\cos36^\circ=\frac{\sqrt{5}+1}{4}$]
If $\tan\theta=\frac{1}{2}$ and $\tan\phi=\frac{1}{3},$ then the value of $\theta+\phi$ is:
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$\frac{\pi}{6}$
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$\pi$
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$0$
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$\frac{\pi}{4}$
The value of $\sin\frac{\pi}{18}+\sin\frac{\pi}{9}+\sin\frac{2\pi}{9}+\sin\frac{5\pi}{18}$ is given by:
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$\sin\frac{7\pi}{18}+\sin\frac{4\pi}{9}$
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$1$
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$\cos\frac{\pi}{6}+\cos\frac{3\pi}{7}$
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$\cos\frac{\pi}{9}+\sin\frac{\pi}{9}$
Which of the following is correct?
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$\sin1^\circ>\sin1$
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$\sin1^\circ<\sin1$
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$\sin1^\circ=\sin1$
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$\sin1^\circ=\frac{\pi}{18^\circ}\sin1$
[Hint: $1\text{radian} =\frac{180^\circ}{\pi}=57^\circ30'\text{approx}$]
$\sin10^\circ$ is greater than $\cos10^\circ.$
View full solution →One value of $\theta$ which satisfies the equation $\sin^4\theta-2\sin^2\theta-1$ lies between 0 and $2\pi.$
View full solution →The equality $\sin\text{A}+\sin2\text{A}+\sin3\text{A}=3$ holds for some real value of A.
View full solution →$\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{8\pi}{15}\cos\frac{16\pi}{15}=\frac{1}{16}$
View full solution →If $\tan(\pi\cos\theta)=\cot(\pi\sin\theta),$ then $\cos\Big(\theta-\frac{\pi}{4}\Big)=\pm\frac{1}{2\sqrt{2}}$
View full solution →If $\text{k}=\sin\Big(\frac{\pi}{18}\Big)\sin\Big(\frac{5\pi}{18}\Big)\sin\Big(\frac{7\pi}{18}\Big),$ then the numerical value of k is _______.
View full solution →The maximum distance of a point on the graph of the function $\text{y}=\sqrt{3}\sin\text{x}+\cos\text{x}$ from x-axis is _______.
View full solution →In a triangle ABC with $\angle\text{C}=90^\circ$ the equation whose roots are tan A and tan B is _______.
[Hint: $\text{A + B}=90^\circ\Rightarrow\tan\text{A}\tan\text{B}=1$ and $\tan\text{A}+\tan\text{B}=\frac{2}{\sin2\text{A}}$ ]
View full solution →$3(\sin\text{x}-\cos\text{x})^4+6(\sin\text{x}+\cos\text{x})^2+4(\sin^6\text{x}+\cos^6\text{x})=$ _______.
View full solution →If $\sin\text{x}+\cos\text{x}=\text{a},$ then, - $\sin^6\text{x}+\cos^6\text{x}=$ _______
- $|\sin\text{x}-\cos\text{x}|=$ _______.
View full solution →If $\tan(\text{A + B})=\text{p},\tan(\text{A}-\text{B})=\text{q},$ then show that $\tan2\text{A}=\frac{\text{p + q}}{1-\text{pq}}$
$\big[$Hint: Use $2\text{A}=(\text{A + B})+(\text{A}-\text{B})\big]$
View full solution →Prove that $4\text{A}=4\sin\text{A}\cos^3\text{A}-4\cos\text{A}\sin^3\text{A}.$
View full solution →If $2\sin^2\theta=3\cos\theta,$ where $0\leq\theta\leq2\pi,$ then find the value of $\theta.$
View full solution →If $\sin\theta+\cos\theta=1,$ then find the general value of $\theta$
View full solution →Find the value of $\tan22^{\circ}30'.$
View full solution →If $\text{a}\cos\theta+\text{b}\sin\theta=\text{m}$ and $\text{a}\sin\theta-\text{b}\cos\theta=\text{n},$ then show that $\text{a}^2+\text{b}^2=\text{m}^2+\text{n}^2.$
View full solution →If $\frac{\sin(\text{x+y})}{\sin(\text{x}-\text{y})}=\frac{\text{a+b}}{\text{a}-\text{b}},$ then show that $\frac{\tan\text{x}}{\tan\text{y}}=\frac{\text{a}}{\text{b}}.$
[Hint: Use Componendo and Dividendo]
View full solution →Prove that $\frac{\tan\text{A}+\sec\text{A}-1}{\tan\text{A}-\sec\text{A}+1}=\frac{1+\sin\text{A}}{\cos\text{A}}$
View full solution →Prove that $\cos\theta\cos\frac{\theta}{2}-\cos3\theta\cos\frac{9\theta}{2}=\sin7\theta\sin8\theta$
$\Big[$Hint: Express $\text{L.H.S.}=\frac{1}{2}\Big[2\cos\theta\cos\frac{\theta}{2}-2\cos3\theta\cos\frac{9\theta}{2}\Big]\Big]$
View full solution →Find the general solution of the equation $\sin\text{x}-3\sin2\text{x}+\sin3\text{x}=\cos\text{x}-3\cos2\text{x}+\cos3\text{x}$
View full solution →If $\theta$ lies in the first quadrant and $\cos\theta=\frac{8}{17},$ then find the value of $\cos(30^\circ+\theta)+\cos(45^\circ-\theta)+\cos(120^\circ-\theta).$
View full solution →Find the value of the expression $\cos^4\frac{\pi}{8}+\cos^4\frac{3\pi}{8}+\cos^4\frac{5\pi}{8}+\cos^4\frac{7\pi}{8}$
[Hint: Simplify the expression to $2\Big(\cos^4\frac{\pi}{8}+\cos^4\frac{3\pi}{8}\Big)=2\Big[\Big(\cos^2\frac{\pi}{8}+\cos^2\frac{3\pi}8{}\Big)^2-2\cos^2\frac{\pi}{8}\cos^2\frac{3\pi}{8}\Big]$
View full solution →In the following match each item given under the column C1 to its correct answer given under the column C2: | | Column C1 | | Column C2 |
| (a) | $\sin(\text{x + y})\sin\text{x}-\text{y}$ | (i) | $\cos^2\text{x}-\sin^2\text{y}$ |
| (b) | $\cos(\text{x + y})\cos(\text{x}-\text{y})$ | (ii) | $\frac{1-\tan\theta}{1+\tan\theta}$ |
| (c) | $\cot\Big(\frac{\pi}{4}+\theta\Big)$ | (iii) | $\frac{1+\tan\theta}{1-\tan\theta}$ |
| (d) | $\tan\Big(\frac{\pi}{4}+\theta\Big)$ | (iv) | $\sin^2\text{x}-\sin^2\text{y}$ |
View full solution →If $\tan\theta=\frac{\sin\alpha-\cos\alpha}{\sin\alpha+\cos\alpha},$ then show that $\sin\alpha+\cos\alpha=\sqrt{2}\cos\theta.$
[Hint: Express $\tan\theta=\tan(\alpha-\frac{\pi}{4})\theta=\alpha-\frac{\pi}{4}$ ]
View full solution →If $\cos(\theta+\phi)=\text{m}\cos(\theta-\phi),$ then prove that $\tan \theta=\frac{1-\text{m}}{1+\text{m}}\cot\phi.$
[Hint: Express $\frac{\cos(\theta+\phi)}{\cos(\theta-\phi)}=\frac{\text{m}}{1}$ and apply Componendo and Dividendo]
View full solution →