- Evaluate the following
(i) sin60° cos30° + sin30° cos 60°
(ii) 2tan2 45° + cos2 30° − sin2 60°
(iii) (cos45°)/(sec30°+cosec30°)
(iv)
(v)
Solution:
(i) sin60° cos30° + sin30° cos 60°
= √3/2 x √3/2 + 1/2 x ½
= 3/4 + 1/4
= 4/4
= 1
(ii) 2tan2 45° + cos2 30° − sin2 60°
= 2(1)2 + 3/4 – 3/4
= 2
(iii) (cos45°)/(sec30°+cosec30°)
(iv)
(v)
2. Choose the correct option and justify your choice.
(i) (2tan30˚)/(1+tan230˚)
(a) sin 60˚
(b) cos 60˚
© tan60˚
(d) sin30˚
(ii) (1-tan245˚)/(1+tan245˚)
(a) tan90˚
(b) 1˚
© sin45˚
(d) 0˚
(iii) sin2A = 2sinA is true when A =
(a) 0˚
(b) 30˚
© 45˚
(d) 60˚
(iv) 2tan30˚/(1-tan230˚)
(a) cos60˚
(b) sin 60˚
©tan60˚
(d) sin30˚
Solution:
(i) (2tan30˚)/(1+tan²30˚)
Out of the given alternatives, only sin60˚= √3/2 .
Hence, (A) is correct.
(ii) (1-tan245˚)/(1+tan245˚)
= [(1-(1)2]/ [(1+(1)2]
= (1-1)/(1+1)
= 0/2
= 0
Hence, (D) is correct.
(iii)sin2A = 2sinA is true when A =
Out of the given alternatives, only A = 0° is correct.
As sin 2A = sin 0° = 0
2sinA = 2sin 0° = 2(0) = 0. Hence, (A) is correct.
(iv) 2tan30˚/(1-tan230˚)
Out of the given alternatives, only tan 60° = √3. Hence, (C) is correct.
3. If tan(A+B) = √3 and tan(A-B) = 1/√3
0˚< A+B ≤ 90˚, A>B, find A and B.
Solution:
tan(A+B) = √3
tan(A+B) = tan60
A+B = 60 ———(1)
tan(A-B) = 1/√3
⇒tan (A − B) = tan30
⇒A − B = 30 ————–(2)
On adding both equations, we obtain 2A = 90
⇒ A = 45
From equation (1), we obtain
45 + B = 60
B = 15
Therefore, ∠A = 45° and ∠B = 15°
4. State whether the following are true or false. Justify your answer.
(i) sin (A + B) = sin A + sin B
(ii) The value of sinθ increases as θ increases
(iii) The value of cos θ increases as θ increases
(iv) sinθ = 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 (A + B) = sin (30° + 60°) = sin 90° = 1
We have, sin A + sin B = sin 30° + sin 60°
1/2 + √3/2 = (1+√3)/2
Clearly, sin (A + B) ≠ sin A + sin B
Hence, the given statement is false.
(ii) The value of sin θ increases as θ increases in the interval of 0° < θ < 90° as sin 0° = 0
sin 30˚= 1/2 = 0.5
sin 45˚= 1/√2 = 0.707
cos60˚= 1/2 = 0.5
cos90˚ = 0
It can be observed that the value of cos θ does not increase in the interval of 0°<θ
<90°. Hence, the given statement is false. >
(iv) sin θ = cos θ for all values of θ.
This is true when θ = 45°
As sin˚45 = 1/√2
cos˚45 = 1/√2
It is not true for all other values of θ.
As sin30˚ = 1/2 and cos30 ˚ = √3/2
Hence, the given statement is false.
(v) cot A is not defined for A = 0°
As CotA= cosA/sinA
cot0˚ = cos0˚/sin0˚ = 1/0 undefined
Hence, the given statement is true.
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