If 2 × sin q × cos(90° - q) = 1/2, where 0° < q < 90°, then find the value of sin(2q) + cos(2q).
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Correct answer: C
Explanation
Has been given :
2 × sinq × cos(90° − q) = 1/2
0° < q < 90°
According to the formula:
cos(90° − q) = sinq
sin2q = 2sinq cosq
2 × sinq × sinq = 1/2
2sin²q = 1/2
sin²q = 1/4
sinq = 1/2
cosq = √3/2
sin2q = 2sinq cosq
= 2 × (1/2) × (√3/2)
= √3/2
cos2q = cos²q − sin²q
= (3/4 − 1/4)
= 1/2
Therefore
sin2q + cos2q
= √3/2 + 1/2 = (√3 + 1)/2