1.

Find the range of 5 cosx – 12 sinx + 7.

Answer»

Given, 5 cosx – 12 sinx + 7 

= 13 (\(\frac{5}{13}cosx - \frac{12}{13}sinx\)) + 7 

Using formula, sin (A + B) = sin A cos B + cos A sin B 

Suppose, sin A = \(\frac{5}{13}\) and cos A = \(\frac{12}{13}\)

= 13 (sin A cosx – cos A sinx) + 7 

= 13 sin (A – x) + 7 

We know the range of sin x is [– 1, 1] 

so, the range of 13 sin (A – x) + 7 is [– 1 × 13 + 7, 1 × 13 + 7] 

= [– 6, 20]



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