Introduction to Trigonometry: 1 mark Questions (CBSE Class 10)

The value of $\sin 90^{\circ }\cos 90^{\circ }-{\sin }^{2}60^{\circ }$ is :

1. (A)$\frac{1}{4}$
2. (B)$\frac{3}{4}$
3. (C)$-\frac{3}{4}$
4. (D)$\frac{5}{4}$

If $\sin \theta =\frac{1}{7}$, then $\tan \theta$ is :

1. (A)$\frac{1}{4\sqrt{3}}$
2. (B)$\frac{1}{2\sqrt{3}}$
3. (C)$\frac{4\sqrt{3}}{7}$
4. (D)$\frac{6}{7}$

The value of $\tan 30^{\circ }\tan 60^{\circ }-\sin 90^{\circ }\cos 90^{\circ }$ is :

1. (A)1
2. (B)0
3. (C)2
4. (D)not defined

If $\sec \theta =\sqrt{3}$, then the value of $\tan \theta$ is :

1. (A)$\sqrt{3}-1$
2. (B)$60^{\circ }$
3. (C)2
4. (D)$\sqrt{2}$

The value of $2\tan 45^{\circ }{\sin }^{2}60^{\circ }-\cos 90^{\circ }$ is :

1. (A)$-\frac{1}{4}$
2. (B)$\frac{3}{2}$
3. (C)$\frac{3}{4}$
4. (D)$\frac{1}{4}$

If value of $\cot \theta$ is $\sqrt{5}$, then $\sin \theta$ equals

1. (A)$\frac{1}{\sqrt{6}}$
2. (B)$\sqrt{6}$
3. (C)$\frac{\sqrt{5}}{6}$
4. (D)$\frac{1}{2}$

Assertion (A) : For an acute angle $\theta$, $\cos \theta$ is always less than 1.
Reason (R) : In a right-angled triangle, hypotenuse is the longest side and $\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}}$.

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

If $\sin \theta =\frac{1}{\sqrt{11}}$, then $\cot \theta$ equals

1. (A)$\frac{\sqrt{11}}{\sqrt{10}}$
2. (B)$\frac{\sqrt{10}}{\sqrt{11}}$
3. (C)$\sqrt{10}$
4. (D)$\sqrt{11}$

Given that $\sin \theta =\frac{a}{b}$, then $\cos \theta$ is equal to :

1. (A)$\frac{b}{\sqrt{b^2-a^2}}$
2. (B)$\frac{b}{a}$
3. (C)$\frac{\sqrt{b^2-a^2}}{b}$
4. (D)$\frac{a}{\sqrt{b^2-a^2}}$

If $\cos A=\frac{1}{2}$, then the value of ${\sin }^{2}A+2{\cos }^{2}A$ is :

1. (A)$\frac{3}{2}$
2. (B)$\frac{5}{4}$
3. (C)$-1$
4. (D)$\frac{1}{2}$

If $\cos y=0$, then what is the value of $\frac{1}{2}\cos \frac{y}{2}$ ?

1. (A)0
2. (B)$\frac{1}{2}$
3. (C)$\frac{1}{\sqrt{2}}$
4. (D)$\frac{1}{2\sqrt{2}}$

If $\cos A=\frac{4}{5}$, then the value of $\tan A$ is :

1. (A)$\frac{3}{5}$
2. (B)$\frac{3}{4}$
3. (C)$\frac{4}{3}$
4. (D)$\frac{5}{3}$

If $2\sin A=1$, then the value of $\tan A+\cot A$ is :

1. (A)$\sqrt{3}$
2. (B)$\frac{4}{\sqrt{3}}$
3. (C)$\frac{\sqrt{3}}{2}$
4. (D)1

Given $\cot \theta =3$, the value of $\cos \theta$ is :

1. (A)$\frac{1}{3}$
2. (B)$\frac{1}{\sqrt{10}}$
3. (C)$\frac{3}{\sqrt{10}}$
4. (D)$\frac{\sqrt{10}}{3}$

Given that $\sin 2\alpha =\frac{\sqrt{3}}{2}$, the value of $\sin 3\alpha$ is :

1. (A)$\frac{3\sqrt{3}}{4}$
2. (B)$\frac{1}{2}$
3. (C)1
4. (D)$\frac{\sqrt{3}}{4}$

The value of $\left(\frac{1}{2}{\tan }^{2}45^{\circ }-{\cos }^{2}60^{\circ }\right)$ is :

1. (A)0
2. (B)$-\frac{1}{2}$
3. (C)$\frac{1}{4}$
4. (D)$-\frac{1}{4}$

Assertion (A) : $\tan 2\theta$ is not defined at $\theta =45^{\circ }$.
Reason (R) : $\sin 90^{\circ }\ne \cos 90^{\circ }$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

The value of $\left(\frac{1}{3}{\cot }^{2}30^{\circ }-\frac{1}{2}{\sec }^{2}60^{\circ }\right)$ is :

1. (A)– 1
2. (B)– 2
3. (C)$\frac{5}{8}$
4. (D)$\frac{7}{8}$

When $\sin A=\frac{1}{3}$, the value of $\cot A$ is

1. (A)$\frac{2\sqrt{2}}{3}$
2. (B)$2\sqrt{2}$
3. (C)$\frac{1}{2\sqrt{2}}$
4. (D)3

$\frac{1+{\tan }^{2}A}{1+{\cot }^{2}A}$ equals to :

1. (A)${\tan }^{2}A$
2. (B)$-1$
3. (C)$-{\tan }^{2}A$
4. (D)${\cot }^{2}A$

If $2\tan A=3$, then value of $\sec A$ equals

1. (A)$\sqrt{\frac{13}{2}}$
2. (B)$\frac{\sqrt{13}}{4}$
3. (C)$\frac{2}{\sqrt{13}}$
4. (D)$\frac{\sqrt{13}}{2}$

$\frac{{\sec }^{2}A-1}{{\sin }^{2}A}$ is same as

1. (A)${\cos }^{2}A$
2. (B)${\sec }^{2}A$
3. (C)$-{\sec }^{2}A$
4. (D)${\cot }^{2}A$

For an acute angle $\theta$, if $\sin \theta =\frac{1}{9}$, then value of $\frac{9cosec\theta +1}{9cosec\theta -1}$ is

1. (A)0
2. (B)$\frac{80}{81}$
3. (C)1
4. (D)$\frac{82}{80}$

Simplest form of $\frac{\sec A}{\sqrt{{\sec }^{2}A-1}}$ is

1. (A)$\sin A$
2. (B)$\tan A$
3. (C)$cosecA$
4. (D)$\cos A$

For an acute angle $\theta$, if $\cos \theta =\frac{1}{8}$, then $\frac{8\sec \theta +1}{8\sec \theta -1}$ equals

1. (A)$\frac{64}{63}$
2. (B)0
3. (C)$\frac{65}{63}$
4. (D)1

Which of the following statements is false ?

1. (A)$\tan 45^{\circ }=\cot 45^{\circ }$
2. (B)$\sin 90^{\circ }=\tan 45^{\circ }$
3. (C)$\sin 30^{\circ }=\cos 30^{\circ }$
4. (D)$\sin 45^{\circ }=\cos 45^{\circ }$

The value of $\left({\tan }^{2}A-\frac{1}{{\cos }^{2}A}\right)$ is :

1. (A)more than 1
2. (B)1
3. (C)0
4. (D)$-1$

The value of $\left({\cot }^{2}A-\frac{1}{{\sin }^{2}A}\right)$ is :

1. (A)more than 1
2. (B)1
3. (C)0
4. (D)$-1$

The value of $\left(\frac{1}{{\sec }^{2}A}+\frac{1}{{\text{cosec}}^{2}A}\right)$ is :

1. (A)more than 1
2. (B)1
3. (C)0
4. (D)$-1$

The value of $\frac{2\tan 60^{\circ }}{1-{\tan }^{2}60^{\circ }}$ is :

1. (A)$-3$
2. (B)$\sqrt{3}$
3. (C)$-\frac{1}{\sqrt{3}}$
4. (D)$-\sqrt{3}$

The value of $({\sin }^{2}A+{\cos }^{2}A)+({\sec }^{2}A-{\tan }^{2}A)-({\cot }^{2}A-{\mathrm{cosec}}^{2}A)$ is :

1. (A)$1$
2. (B)$-1$
3. (C)$3$
4. (D)$-2$

$(\sin \theta +\cos \theta {)}^2+(\sin \theta -\cos \theta {)}^2=$

1. (A)$1$
2. (B)$2$
3. (C)$2+2\sin \theta \cos \theta$
4. (D)$2+4\sin \theta \cos \theta$

$(\sec \theta -\cos \theta {)}^2+{\sin }^2\theta -{\tan }^2\theta =?$

1. (A)$0$
2. (B)$1$
3. (C)$2$
4. (D)$4$

The value of $\frac{{\sec }^{2}30^{\circ }+{\tan }^{2}30^{\circ }}{{\sin }^{2}45^{\circ }+{\cos }^{2}45^{\circ }}$ is :

1. (A)1
2. (B)$\frac{5}{3}$
3. (C)$\frac{13}{3}$
4. (D)7

Assertion (A) : For $\sin \theta =1$, $\cos \theta$ must be 0.
Reason (R) : ${\sin }^2\theta -{\cos }^2\theta =1$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

The value of $\frac{{\cot }^{2}A-{\text{cosec}}^{2}A}{\sin 30^{\circ }+\cos 60^{\circ }}$ is :

1. (A)1
2. (B)$-1$
3. (C)$\frac{2}{1+\sqrt{3}}$
4. (D)$\frac{-2}{1+\sqrt{3}}$

Assertion (A) : For an acute angle $\theta$, $\cot \theta =1\Rightarrow \text{cosec }\theta =2$.
Reason (R) : ${\text{cosec}}^2\theta -{\cot }^2\theta =1$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

The value of $\theta$ for which $\sin 2\theta =\tan 45^{\circ }$ is :

1. (A)$22.5^{\circ }$
2. (B)$30^{\circ }$
3. (C)$45^{\circ }$
4. (D)$90^{\circ }$

Assertion (A) : For an angle $\theta$, $\sec \theta =1\Rightarrow \tan \theta =0$.
Reason (R) : ${\sec }^2\theta +{\tan }^2\theta =1$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

If $\sqrt{2}\sin \theta =1$, then $\cot \theta \times \text{cosec}\theta$ is equal to :

1. (A)$\frac{1}{\sqrt{2}}$
2. (B)$\frac{1}{2\sqrt{2}}$
3. (C)$\sqrt{2}$
4. (D)$\frac{1}{2}$

If $\sec \theta -\tan \theta =2$, then $\sec \theta +\tan \theta$ is equal to :

1. (A)$\frac{1}{2}$
2. (B)$\sqrt{2}$
3. (C)$\frac{1}{\sqrt{2}}$
4. (D)2

If $\sin A=\frac{2}{3}$, then $\cos A$ is equal to :

1. (A)$\frac{3}{2}$
2. (B)$\frac{\sqrt{5}}{3}$
3. (C)$\frac{1}{3}$
4. (D)$\frac{1}{\sqrt{3}}$

If $\tan A=\frac{1}{2}$, then $\sin A$ is equal to :

1. (A)$\frac{2}{\sqrt{5}}$
2. (B)$\frac{1}{\sqrt{3}}$
3. (C)$\frac{1}{\sqrt{5}}$
4. (D)1

If tan A = 1, then 3 sin A + cos A is equal to

1. (A)$4\sqrt{2}$
2. (B)4
3. (C)$2\sqrt{2}$
4. (D)$4\times 45^{\circ }$

Assertion (A) : In a right angle triangle ABC, $\angle B=90^{\circ }$. Therefore the value of cos (A + C) is equal to 0.
Reason (R) : $A+B+C=180^{\circ }$ and $\cos 90^{\circ }=0$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

If sin A = cos A, then $\frac{1-\tan A}{1+\tan A}$ is equal to

1. (A)1
2. (B)–1
3. (C)0
4. (D)not defined

If $\sqrt{3}\sin \theta =\cos \theta$, then value of $\theta$ is

1. (A)$\sqrt{3}$
2. (B)$60^{\circ }$
3. (C)$\frac{1}{\sqrt{3}}$
4. (D)$30^{\circ }$

In $△ABC$, $\angle B=90^{\circ }$. If $\frac{AB}{AC}=\frac{1}{2}$, then cos C is equal to

1. (A)$\frac{3}{2}$
2. (B)$\frac{1}{2}$
3. (C)$\frac{\sqrt{3}}{2}$
4. (D)$\frac{1}{\sqrt{3}}$

If $\sin \theta =\frac{1}{9}$, then $\tan \theta$ is equal to

1. (A)$\frac{1}{4\sqrt{5}}$
2. (B)$\frac{4\sqrt{5}}{9}$
3. (C)$\frac{1}{8}$
4. (D)$4\sqrt{5}$

If $\theta$ is an acute angle and $7+4\sin \theta =9$, then the value of $\theta$ is :

1. (A)$90^{\circ }$
2. (B)$30^{\circ }$
3. (C)$45^{\circ }$
4. (D)$60^{\circ }$

The value of ${\tan }^2\theta -\left(\frac{1}{\cos \theta }\times \sec \theta \right)$ is :

1. (A)$1$
2. (B)$0$
3. (C)$-1$
4. (D)$2$

If $\alpha +\beta =90^{\circ }$ and $\alpha =2\beta$, then ${\cos }^2\alpha +{\sin }^2\beta$ is equal to :

1. (A)$0$
2. (B)$\frac{1}{2}$
3. (C)$1$
4. (D)$2$

The value of $(\tan A\mathrm{cosec}A{)}^2-(\sin A\sec A{)}^2$ is :

1. (A)$0$
2. (B)$1$
3. (C)$-1$
4. (D)$2$

If $7{\cos }^2\theta +3{\sin }^2\theta =4$, then the value of $\theta$ is :

1. (A)$30^{\circ }$
2. (B)$45^{\circ }$
3. (C)$60^{\circ }$
4. (D)$90^{\circ }$

If $\sin 30^{\circ }\tan 45^{\circ }=\frac{\sec 60^{\circ }}{k}$, then the value of k is :

1. (A)4
2. (B)3
3. (C)2
4. (D)1

If $\sin (\alpha +\beta )=1$, then the value of $\sin \left(\frac{\alpha +\beta }{2}\right)$ is :

1. (A)$\frac{1}{\sqrt{2}}$
2. (B)$\frac{1}{2}$
3. (C)0
4. (D)1

If $\sin \theta =\cos \theta$ $(0^{\circ }<\theta <90^{\circ })$, then the value of $\sec \theta \cdot \sin \theta$ is :

1. (A)$\frac{1}{\sqrt{2}}$
2. (B)$\sqrt{2}$
3. (C)0
4. (D)1

If $\tan 3\theta =\sqrt{3}$, then $\frac{\theta }{2}$ equals :

1. (A)$60^{\circ }$
2. (B)$30^{\circ }$
3. (C)$20^{\circ }$
4. (D)$10^{\circ }$

$(\cot \theta +\tan \theta )$ equals :

1. (A)$\text{cosec}\theta \sec \theta$
2. (B)$\sin \theta \sec \theta$
3. (C)$\cos \theta \tan \theta$
4. (D)$\sin \theta \cos \theta$

If $\sin 4\theta =\frac{\sqrt{3}}{2}$, then $\frac{\theta }{3}$ equals :

1. (A)$60^{\circ }$
2. (B)$20^{\circ }$
3. (C)$15^{\circ }$
4. (D)$5^{\circ }$

$\frac{\cos \theta }{\sqrt{1-{\cos }^2\theta }}$ is equal to :

1. (A)$\cot \theta$
2. (B)$\sqrt{\cos \theta }$
3. (C)$\frac{\cos \theta }{\sqrt{\sin \theta }}$
4. (D)$\tan \theta$

$\tan 2A=3\tan A$ is true, when the measure of $\angle A$ is :

1. (A)$90^{\circ }$
2. (B)$60^{\circ }$
3. (C)$45^{\circ }$
4. (D)$30^{\circ }$

Which of the following statements is true ?

1. (A)$\sin 20^{\circ }>\sin 70^{\circ }$
2. (B)$\sin 20^{\circ }>\cos 20^{\circ }$
3. (C)$\cos 20^{\circ }>\cos 70^{\circ }$
4. (D)$\tan 20^{\circ }>\tan 70^{\circ }$

Which of the following is a trigonometric identity ?

1. (A)${\sin }^2\theta =1+{\cos }^2\theta$
2. (B)${\text{cosec}}^2\theta +{\cot }^2\theta =1$
3. (C)${\sec }^2\theta =1+{\tan }^2\theta$
4. (D)$\sin 2\theta =2\sin \theta$

$\sec A=2\cos A$ is true for $A=$

1. (A)$0^{\circ }$
2. (B)$30^{\circ }$
3. (C)$45^{\circ }$
4. (D)$60^{\circ }$

$\frac{1-{\tan }^{2}30^{\circ }}{1+{\tan }^{2}30^{\circ }}$ is equal to

1. (A)$\sin 60^{\circ }$
2. (B)$\cos 60^{\circ }$
3. (C)$\tan 60^{\circ }$
4. (D)$\sec 60^{\circ }$