Chapter 8 · Trigonometry

NCERT Class 10 Maths Exercise 8.1

All 11 questions of NCERT Class 10 Maths Exercise 8.1 (Trigonometry), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   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
   In ​Fig. ​8.13, ​find $\tan P-\cot R$.

   Show answer

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

   Step-by-step solution →
3. Q3
   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
   Given 15 cot ​A ​= ​8, find ​sin ​A and ​sec ​A.
   Show answer

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

   Step-by-step solution →
5. Q5
   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
   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.

   Step-by-step solution →
7. Q7
   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
   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.

   Step-by-step solution →
9. Q9
   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
    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
    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 →