NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry
Table of Contents
Exercise 8.1
Question 1:
In \(\triangle\) ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine :
(i) sin A, cos A
(ii) sin C, cos C
Question 2:
In Fig. 8.13, find tan P – cot R.
Question 3:
If sin A = \(\frac{3}{4}\), calculate cos A and tan A.
Question 4:
Given 15 cot A = 8, find sin A and sec A.
Question 5:
Given sec \(\theta\) = \(\frac{13}{12}\) , calculate all other trigonometric ratios.
Question 6:
If \(\angle\) A and \(\angle\) B are acute angles such that cos A = cos B, then show that \(\angle\) A = \(\angle\) B.
Question 7:
If \(\cot {\theta}\) = \(\frac{7}{8}\), evaluate: (i) \(\frac{(1 + \sin {\theta})(1 – \sin {\theta})}{(1 + \cos {\theta})(1- \cos {\theta})}\), (ii) 1 + \(\cot^2 {\theta}\)
Question 8:
If 3 cot A = 4, check whether \(\frac{1 – \tan^2 A}{1 + \tan^2 A}\) = \(\cos^2 A\) – \(\sin^2 A\) or not.
Question 9:
In triangle ABC, right-angled at B, if tan A = \(\frac{1}{\sqrt{3}}\), find the value of:
(i) sin A cos C + cos A sin C
(ii) cos A cos C – sin A sin C
Question 10:
In \(\triangle\) PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.
Question 11:
State whether the following are true or false. Justify your answer.
(i) The value of tan A is always less than 1.
(ii) sec A = \(\frac{12}{5}\) for some value of angle A.
(iii) cos A is the abbreviation used for the cosecant of angle A.
(iv) cot A is the product of cot and A.
(v) sin \(\theta\) = \(\frac{4}{3}\) for some angle \(\theta\).
Exercise 8.2
Question 1:
Evaluate the following :
(i) sin 60° cos 30° + sin 30° cos 60°
(ii) 2 tan2 45° + cos2 30° – sin2 60°
(iii) \(\frac{\cos {45°}}{\sec {30°} + cosec {30°}}\)
(iv) \(\frac{\sin {30°} + \tan {45° – cosec {60°}}}{\sec {30°} + \cos {60°} + \cot {45°}}\)
(v) \(\frac{5\cos^2 {60°} + 4\sec^2 {30°} – \tan^2 {45°}}{\sin^2 {30°} + \cos^2 {30°}}\)
Solution
(i) sin 60° cos 30° + sin 30° cos 60°
\(\quad\) = \(\frac{\sqrt{3}}{2} \times\) \(\frac{\sqrt{3}}{2}\) + \(\frac{1}{2} \times\) \(\frac{1}{2}\)
\(\quad\) = \(\frac{3}{4} \) + \(\frac{1}{4}\)
\(\quad\) = \(\frac{4}{4}\) = 1
(ii) 2 tan2 45° + cos2 30° – sin2 60°
\(\quad\) = 2 (1)2 + \((\frac{\sqrt{3}}{2})^2\) – \((\frac{\sqrt{3}}{2})^2\)
\(\quad\) = 2
(iii) \(\frac{\cos {45°}}{\sec {30°} + cosec {30°}}\)
\(\quad\) = \(\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}} + 2}\)
\(\quad\) = \(\frac{\frac{1}{\sqrt{2}}}{\frac{2 + 2\sqrt{3}}{\sqrt{3}}}\)
\(\quad\) = \(\frac{1}{\sqrt{2}}\) \(\times\) \(\frac{\sqrt{3}}{2 + 2\sqrt{3}}\)
(iv) \(\frac{\sin {30°} + \tan {45° – cosec {60°}}}{\sec {30°} + \cos {60°} + \cot {45°}}\)
=\(\frac{\frac{1}{2} + 1 – \frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}} – \frac{1}{2} + 1}\)
= \(\frac{\frac{\sqrt{3} + 2\sqrt{3} – 4}{2\sqrt{3}}}{\frac{4 + \sqrt{3} + 2\sqrt{3}}{2\sqrt{3}}}\)
= \(\frac{\sqrt{3} + 2\sqrt{3} – 4}{4 + \sqrt{3} + 2\sqrt{3}}\)
=\(\frac{3\sqrt{3} – 4}{3\sqrt{3} + 4}\) \(\times\) \(\frac{3\sqrt{3} – 4}{3\sqrt{3} – 4}\)
= \(\frac{(3\sqrt{3} – 4)^2}{(3\sqrt{3} + 4)(3\sqrt{3} – 4)}\)
= \(\frac{27 + 16 – 24\sqrt{3}}{27 -16}\)
= \(\frac{43 – 24\sqrt{3}}{11}\)
(v) \(\frac{5\cos^2 {60°} + 4\sec^2 {30°} – \tan^2 {45°}}{\sin^2 {30°} + \cos^2 {30°}}\)
= \(\frac{5()^2 + 4()^2 – (1)^2}{(\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2}\)
Question 2:
Choose the correct option and justify your choice :
(i) \(\frac{2 \tan {30^o}}{1 + \tan^2 {30^o}}\) =
\(\quad\) (A) \(\sin {60^o}\) (B) \(\cos {60^0}\) (C) \(\tan {60^0}\) (D) \(\sin {30^0}\)
Question 3:
If tan (A + B) = \(\sqrt{3}\) and tan (A – B) = \(\frac{1}{\sqrt3}\); 0° < A + B \(\le\) 90°; A > B, find A and B.
Solution
tan (A + B) = 3 \(\Rightarrow\)
and tan (A – B) = \(\frac{1}{\sqrt3}\)
Question 4:
State whether the following are true or false. Justify your answer.
(i) sin (A + B) = sin A + sin B.
(ii) The value of sin \(\theta\) increases as \(\theta\) increases.
(iii) The value of cos \(\theta\) increases as \(\theta\) increases.
(iv) sin \(\theta\) = cos \(\theta\) for all values of \(\theta\).
(v) cot A is not defined for A = 0°.
Exercise 8.3
Question 1:
Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.
Question 2:
Write all the other trigonometric ratios of \(\angle\) A in terms of sec A.
Question 3:
Choose the correct option. Justify your choice.
(i). \( 9 \sec^2 A – 9 \tan^2 A = \)
(A) \( 1 \) (B) \( 9 \) (C) \( 8 \) (D) \( 0 \)
(ii). \( (1 + \tan \theta + \sec \theta)(1 + \cot \theta – \csc \theta) = \)
(A) \( 0 \) (B) \( 1 \) (C) \( 2 \) (D) \( -1 \)
(iii). \( (\sec A + \tan A)(1 – \sin A) = \)
(A) \( \sec A \) (B) \( \sin A \) (C) \( \csc A \) (D) \( \cos A \)
(iv). \( \frac{1 + \tan^2 A}{1 + \cot^2 A} = \)
(A) \( \sec^2 A \) (B) \( -1 \) (C) \( \cot^2 A \) (D) \( \tan^2 A \)
Question 4:
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
