sin²60° - tan60° by sin²30° + cosec²30°
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Exercise - 8.2
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Introduction to Trigonometry
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) ncert mathSolution :-
ncert math
(iv) trigonometorySolution :-
trigonometory
(v) trigonometorySolution :-
trigonometory
Question-2 :- Choose the correct option and justify your choice : (i)ncert math(A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30°
Solution :-
ncert math
Therefore, sin 60° = √3/2.
So, Option A is correct Answer.
(ii) ncert math(A) tan 90° (B) 1 (C) sin 45° (D) 0
Solution :-
ncert math
So, Option D is correct Answer.
(iii) sin 2A = 2 sin A is true when A =
(A) 0° (B) 30° (C) 45° (D) 60°
Solution :-
sin 2A = sin 0° = 0
2 sin A = 2 sin 0° = 2 x 0 = 0
So, Option A is correct Answer.
(iv) ncert math(A) cos 60° (B) sin 60° (C) tan 60° (D) sin 30°
Solution :-
ncert math
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°.
Solution :-
(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.
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Question :- Sin²60° - tan60° by sin²30° + cosec²30°
Answer is attached
