Correct Answer - Option 3 : 12 and 13
Trigonometry identity used:
Sin2θ + cos2θ = 1
Calculation:
12 sin2θ + 13 cos2θ
= 12 sin2θ + 12 cos2θ + cos2θ
= 12 (sin2θ + cos2θ) + cos2θ
= 12 + cos2θ
For minimum value,
Minimum value of cosθ = –1
But cos2θ ≥ 0, when θ = 90°
So, cos0° = 1,
Then, required minimum value
= 12 + 0
= 12.
For the maximum value,
Maximum value of cosθ = 1
And cos2θ =1
Then, required maximum value,
= 12 + 1 = 13
∴The minimum and maximum values of 12 sin2θ +13 cos2θ are 12 and 13.