Class 10 Maths · Chapter 8

Introduction to Trigonometry

108 Class 10 Maths questions on Introduction to Trigonometry from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 1 of 6

1. Q1 NCERT · Exercise 8.1
   In $△ABC$, ​right-angled ​at B, AB = 24 cm, ​BC ​= 7 cm. ​Determine: ​

   1. (i)$\sin A,\cos A$ ​
   2. (ii)$\sin C,\cos C$
   Show answer

   (i) $\sin A=\frac{7}{25},\cos A=\frac{24}{25}$; (ii) $\sin C=\frac{24}{25},\cos C=\frac{7}{25}$.

   Step-by-step solution →
2. Q2 NCERT · Exercise 8.1
   In ​Fig. ​8.13, ​find $\tan P-\cot R$.

   Show answer

   $\tan P-\cot R=0$.

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3. Q3 NCERT · Exercise 8.1
   If $\sin A=\frac{3}{4}$, calculate $\cos A$ ​and $\tan A$.
   Show answer

   $\cos A=\frac{\sqrt{7}}{4}$ and $\tan A=\frac{3}{\sqrt{7}}=\frac{3\sqrt{7}}{7}$.

   Step-by-step solution →
4. Q4 NCERT · Exercise 8.1
   Given 15 cot ​A ​= ​8, find ​sin ​A and ​sec ​A.
   Show answer

   $\sin A=\frac{15}{17}$ and $\sec A=\frac{17}{8}$.

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5. Q5 NCERT · Exercise 8.1
   Given $\sec \theta =\frac{13}{12}$, ​calculate ​all other ​trigonometric ​ratios.
   Show answer

   $\cos \theta =\frac{12}{13}$, $\sin \theta =\frac{5}{13}$, $\tan \theta =\frac{5}{12}$, $\csc \theta =\frac{13}{5}$, $\cot \theta =\frac{12}{5}$.

   Step-by-step solution →
6. Q6 NCERT · Exercise 8.1
   If ​angle A and ​angle B are acute ​angles ​such that cos A = ​cos B, then show ​that angle A = ​angle B.
   Show answer

   Since $\cos$ is strictly decreasing (one-to-one) on $0^{\circ }$ to $90^{\circ }$, equal cosines of acute angles force $A=B$. Hence proved.

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7. Q7 NCERT · Exercise 8.1
   If $\cot \theta =\frac{7}{8}$, ​evaluate: ​

   1. (i)$\frac{(1+\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(1-\cos \theta )}$, ​
   2. (ii)${\cot }^2\theta$.
   Show answer

   (i) $\frac{49}{64}$; (ii) $\frac{49}{64}$.

   Step-by-step solution →
8. Q8 NCERT · Exercise 8.1
   If $3\cot A=4$, ​check ​whether $\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}={\cos }^{2}A-{\sin }^{2}A$ or ​not.
   Show answer

   Yes. Both sides equal $\frac{7}{25}$, so $\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}={\cos }^{2}A-{\sin }^{2}A$ holds.

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9. Q9 NCERT · Exercise 8.1
   In triangle ​ABC, ​right-angled ​at ​B, if $\tan A=\frac{1}{\sqrt{3}}$, find ​the ​value of:

   1. (i)$\sin A\cos C+\cos A\sin C$
   2. (ii)$\cos A\cos C-\sin A\sin C$
   Show answer

   (i) $1$; (ii) $0$.

   Step-by-step solution →
10. Q10 NCERT · Exercise 8.1
    In $\Delta PQR$, right-angled ​at $Q$, $PR+QR=25$ ​cm ​and $PQ=5$ cm. ​Determine the ​values of $\sin P,\cos P$ and $\tan P$.
    Show answer

    $\sin P=\frac{12}{13}$, $\cos P=\frac{5}{13}$, $\tan P=\frac{12}{5}$.

    Step-by-step solution →
11. Q11 NCERT · Exercise 8.1
    State whether the following are true or false. Justify your answer.

    1. (i)​The value of $\tan A$ is always less than $1$.
    2. (ii)$\sec A=\frac{12}{5}$ for ​some value of angle $A$.
    3. (iii)$\cos A$ is the abbreviation used for the cosecant of angle $A$.
    4. (iv)$\cot A$ is the product of $\cot$ ​and $A$. ​
    5. (v)$\sin \theta =\frac{4}{3}$ ​for some ​angle $\theta$.
    Show answer

    (i) False; (ii) True; (iii) False; (iv) False; (v) False.

    Step-by-step solution →
12. Q12 NCERT · Exercise 8.2
    Evaluate ​the ​following: ​

    1. (i)$\sin 60^{\circ }\cos 30^{\circ }+\sin 30^{\circ }\cos 60^{\circ }$ ​
    2. (ii)$2{\tan }^{2}45^{\circ }+{\cos }^{2}30^{\circ }-{\sin }^{2}60^{\circ }$ ​
    3. (iii)$\frac{\cos 45^{\circ }}{\sec 30^{\circ }+\csc 30^{\circ }}$ ​
    4. (iv)$\frac{\sin 30^{\circ }+\tan 45^{\circ }-\csc 60^{\circ }}{\sec 30^{\circ }+\cos 60^{\circ }+\cot 45^{\circ }}$
    5. (v)$\frac{5{\cos }^{2}60^{\circ }+4{\sec }^{2}30^{\circ }-{\tan }^{2}45^{\circ }}{{\sin }^{2}30^{\circ }+{\cos }^{2}30^{\circ }}$.
    Show answer

    (i) $1$; (ii) $2$; (iii) $\frac{3\sqrt{2}-\sqrt{6}}{8}$; (iv) $\frac{43-24\sqrt{3}}{11}$; (v) $\frac{67}{12}$.

    Step-by-step solution →
13. Q13 NCERT · Exercise 8.2
    Choose the correct option ​and justify ​your choice:

    1. (i)$\frac{2\tan 30^{\circ }}{1+{\tan }^{2}30^{\circ }}=$ ​(A) $\sin 60^{\circ }$ ​(B) $\cos 60^{\circ }$ (C) $\tan 60^{\circ }$ ​(D) $\sin 30^{\circ }$. ​
    2. (ii)$\frac{1-{\tan }^{2}45^{\circ }}{1+{\tan }^{2}45^{\circ }}=$ (A) $\tan 90^{\circ }$ (B) $1$ (C) $\sin 45^{\circ }$ (D) $0$.
    3. (iii)$\sin 2A=2\sin A$ is ​true when $A=$ (A) $0^{\circ }$ (B) $30^{\circ }$ ​(C) $45^{\circ }$ (D) $60^{\circ }$.
    4. (iv)$\frac{2\tan 30^{\circ }}{1-{\tan }^{2}30^{\circ }}=$ (A) $\cos 60^{\circ }$ (B) $\sin 60^{\circ }$ (C) $\tan 60^{\circ }$ (D) $\sin 30^{\circ }$.
    Show answer

    (i) (A) $\sin 60^{\circ }$; (ii) (D) $0$; (iii) (A) $0^{\circ }$; (iv) (C) $\tan 60^{\circ }$.

    Step-by-step solution →
14. Q14 NCERT · Exercise 8.2
    If $\tan (A+B)=\sqrt{3}$ ​and $\tan (A-B)=\frac{1}{\sqrt{3}}$; $0^{\circ }<A+B\le 90^{\circ }$; $A>B$, find $A$ ​and $B$.
    Show answer

    $A=45^{\circ }$ and $B=15^{\circ }$.

    Step-by-step solution →
15. Q15 NCERT · Exercise 8.2
    State whether the ​following are ​true ​or false. Justify your answer.

    1. (i)$\sin (A+B)=\sin A+\sin B$.
    2. (ii)The value of $\sin \theta$ increases as $\theta$ increases.
    3. (iii)The value of $\cos \theta$ ​increases as $\theta$ ​increases.
    4. (iv)$\sin \theta =\cos \theta$ for ​all values of $\theta$.
    5. (v)$\cot A$ is not defined for $A=0^{\circ }$.
    Show answer

    (i) False; (ii) True; (iii) False; (iv) False; (v) True.

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16. Q16 Exemplar · Exercise 8.1
    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)$\frac{5}{3}$
    Show answer

    (b) $\frac{3}{4}$

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17. Q17 Exemplar · Exercise 8.1
    If $\sin A=\frac{1}{2}$, ​then ​the ​value of $\cot A$ ​is

    1. (a)$\sqrt{3}$
    2. (b)$\frac{1}{\sqrt{3}}$
    3. (c)$\frac{\sqrt{3}}{2}$
    4. (d)$1$
    Show answer

    (a) $\sqrt{3}$

    Step-by-step solution →
18. Q18 Exemplar · Exercise 8.1
    The value ​of ​the expression $[\csc (75^{\circ }+\theta )-\sec (15^{\circ }-\theta )-\tan (55^{\circ }+\theta )+\cot (35^{\circ }-\theta )]$ ​is

    1. (a)$-1$
    2. (b)$0$
    3. (c)$1$
    4. (d)$\frac{3}{2}$
    Show answer

    (b) $0$

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19. Q19 Exemplar · Exercise 8.1
    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)$\frac{a}{\sqrt{b^2-a^2}}$
    Show answer

    (c) $\frac{\sqrt{b^2-a^2}}{b}$

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20. Q20 Exemplar · Exercise 8.1
    If $\cos (\alpha +\beta )=0$, ​then $\sin (\alpha -\beta )$ ​can ​be reduced ​to

    1. (a)$\cos \beta$
    2. (b)$\cos 2\beta$
    3. (c)$\sin \alpha$
    4. (d)$\sin 2\alpha$
    Show answer

    (b) $\cos 2\beta$

    Step-by-step solution →