Question-1 :- Evaluate: (i) sin 60° cos 30° + sin 30° cos 60°
Solution :-sin 60° cos 30° + sin 30° cos 60° = √3/2 x √3/2 + 1/2 x 1/2 = 3/4 + 1/4 = (3 + 1)/4 = 4/4 = 1
(ii) 2 tan² 45° + cos² 30° – sin² 60°
Solution :-2 tan² 45° + cos² 30° – sin² 60° = 2 x 1 + (√3/2)² - (√3/2)² = 2 + 3/4 - 3/4 = 2
(iii)
(iv)
(v)
Question-2 :-
Choose the correct option and justify your choice : (i)
(A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30°
Therefore, sin 60° = √3/2. So, Option A is correct Answer.
(ii)
(A) tan 90° (B) 1 (C) sin 45° (D) 0
So, Option D is correct Answer.
(iii)
sin 2A = 2 sin A is true when A =
(A) 0° (B) 30° (C) 45° (D) 60°
sin 2A = sin 0° = 0 2 sin A = 2 sin 0° = 2 x 0 = 0 So, Option A is correct Answer.
(iv)
(A) cos 60° (B) sin 60° (C) tan 60° (D) sin 30°
Therefore, tan 60° = √3/2. So, Option C is correct Answer.
Question-3 :- If tan (A + B) = √3 and tan (A – B) = 1/√3; 0° < A + B ≤ 90°; A > B, find A and B.
Solution :-Since, tan (A + B) = √3, tan (A + B) = tan 60° Therefore, A + B = 60° .....(1) Also, since tan (A – B) = 1/√3, tan (A – B) = tan 30° Therefore, A - B = 30° ......(2) Solving (1) and (2), A + B + A - B = 60° + 30° 2A = 90° A = 90°/2 A = 45° Put in (1) equation A - B = 30° 45° - B = 30° -B = 30° - 45° -B = -15° B = 15° we get : A = 45° and B = 15°.
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 θ increases as θ increases.
(iii) The value of cos θ increases as θ increases.
(iv) sin θ = cos θ for all values of θ.
(v) cot A is not defined for A = 0°.
(i) sin (A + B) = sin A + sin B. Let A = 30° and B = 60° sin(30° + 60°) = sin 90° = 1 sin 30° + sin 60° = 1/2 + √3/2 = (1 + √3)/2 So, statement is not equal and it is false statement.
(ii) The value of sin θ increases as θ increases. sin 0° = 0 sin 30° = 1/2 = 0.5 sin 45° = 1/√2 = 0.7 sin θ increases as θ increases So, this statement is true.
(iii) The value of cos θ increases as θ increases. cos 0° = 1 cos 30° = √3/2 = 0.8 cos 45° = 1/√2 = 0.7 Therefore, cos θ decreases as θ increases. So, this statement is false.
(iv) sin θ = cos θ for all values of θ. sin 0° = 0, cos 0° = 1 sin 30° = 1/2, cos 30° = √3/2 sin 45° = 1/√2, cos 45° = 1/√2 sin 60° = √3/2, cos 60° = 1/2 So, this statement is false.
(v) cot A is not defined for A = 0°. cot A = cos A/sin A cot 0° = cos 0°/sin 0° cot 0° = 1/0 = not defined So, this statement is true.