EXERCISE 8.1
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
2. In Fig. 8.13, find tan P – cot R.
3. If sin A = \(\frac{3}{4}\) , Calculate cos A and tan A.
4. Given 15 cot A = 8, find sin A and sec A.
5. Given sec q = \(\Large{\frac{13}{12}}\), calculate all other trigonometric ratios.
6. If \(\angle \) A and \(\angle \) B are acute angles such that cos A = cos B, then show that \(\angle \) A = \(\angle \) B.
7. If cot q = \(\Large{\frac{7}{8}}\), evaluate : (i)\(\Large{ \frac{(1+ sin \theta )(1-sin \theta)}{(1+ cos\theta) (1- cos \theta)}} \) (ii) \(cot^2 \theta \)
8. If 3 cot A = 4, check whether \(\Large{ \frac{1 – tan ^2 A }{1 + tan ^2 A }} = cos^2 A – sin^2A \) or not.
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
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.
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 = \(\large{\frac{12}{5}}\), 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 q = \(\large{\frac{4}{3}}\), for some angle q.
EXERCISE 8.2
1. 1. Evaluate the following :
(i) sin 60° cos 30° + sin 30° cos 60°
(ii) \(2 tan^2 45° + cos^2 30° – sin^2 60° \)
(iii) \(\Large{\frac{\textup{cos 45°}}{\textup{sec 30° + cosec 30°}}}\)
(iv) \(\Large{\frac{\textup{sin 30° + tan 45° – cosec 60°}}{\textup{sec 30° + cos 60° + cot 45°}}}\)
(v) \(\Large{\frac{5 cos^2 60° + 4 sec^2 30° tan^2 45°}{sin^2 30° + cos^2 30° }}\)
2. Choose the correct option and justify your choice :
(i) \(\Large{\frac{2 tan 30°}{ 1+ tan^2 30° }}\) =
(A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30°
(ii) \(\Large{\frac{1 – tan^2 45°}{ 1 + tan^2 45° }}\) =
(A) tan 90° (B) 1 (C) sin 45° (D) 0
(iii) sin 2A = 2 sin A is true when A =
(A) 0° (B) 30° (C) 45° (D) 60°
(iv) \(\Large{\frac{2 tan 30°}{ 1- tan^2 30° }}\) =
(A) cos 60° (B) sin 60° (C) tan 60° (D) sin 30°
3. If tan (A + B) = \(\sqrt{3}\) and tan (A – B) = \(\frac{1}{\sqrt{3}}\); 0° < A + B \(\le\) 90°; A > B, find A and B.
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 [OMIT for 2022/23]
EXERCISE 8.4
1. Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.
2. Write all the other trigonometric ratios of Ð A in terms of sec A.
3. Evaluate :
(i) \(\Large{\frac{ sin^2 63° + sin^2 27°}{ sin^2 17° + sin^2 73° }}\) [OMIT]
(ii) sin 25° cos 65° + cos 25° sin 65° [OMIT]
4. Choose the correct option. Justify your choice.
(i) \( 9 sec^2 A – 9 tan^2 A\) =
\( (A) 1 \qquad\;(B) 9 \qquad\;(C) 8 \qquad\;(D) 0 \)
(ii) \((1 + tan \theta + sec \theta) (1 + cot \theta – cosec \theta) \) =
\( (A) tan 90° \qquad\;(B) 1 \qquad\;(C) sin 45° \qquad\;(D) 0 \)
(iii) (sec A + tan A) (1 – sin A) =
(A) sec A (B) sin A (C) cosec A (D) cos A
(iv) \(\Large{\frac{1 + tan^2 A}{ 1 + cot^2 A }}\) =
(A) \(sec^2 A (B) –1 (C) cot^2 A (D) tan^2 A \)
5. Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(i) \( (cosec \theta – cot \theta)^2 \) = \(\Large{\frac{1-cos \theta}{1 + cos \theta}} \)(ii) \(\Large{\frac{cos A}{1 + sin A } + \frac{1+ sin A }{ cos A }}\) = 2 Sec A
(iii) \(\Large{\frac{tan \theta }{1 – cot \theta } + \frac{ cot \theta }{ 1- tan \theta }} = 1 + sec \theta cosec \theta \)
[Hint : Write the expression in terms of \(sin \theta\) and cos \(\theta \) ]
(iv) \(\Large{\frac{1 + sec A }{ sec A } = \frac{ sin ^A }{ 1- cos A }} \)
[Hint : Simplify LHS and RHS separately]
(v) \(\Large{\frac{cos A – sin A + 1}{ cos A + sin A – 1 } = cosec A + cot A}\), using the identity \(cosec^2 A = 1 + cot^2 A. \)
(vi) \(\large{ \sqrt{\frac{1+ sin A}{1 – sin A}}} = sec A + tan A \)
(vii) \(\Large{ \frac{sin \theta -2 sin^3 \theta}{2cos^3 \theta – cos \theta}} = tan \theta \)
(viii) \((sin A + cosec A)^2 + (cos A + sec A)^2 = 7 + tan^2 A + cot^2 A \)
(ix) (cosec A – sin A) (sec A – cos A) = \(\large{\frac{1}{tan A + cot A}}\)
(x) \( \Large{\left( \frac{1+ tan^2 A}{1+ cot^2 A} \right) = \left( {\frac{1- tanA}{1- cotA}} \right)^2 = tan^2 A }\)