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in Trigonometry by (33.4k points)
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Find the maximum and minimum value of 7 cos x + 24 sin x.

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y = 7 cos x + 24 sin x

y2 = (7 cos x + 24 sinx)2

= 49 cos2x + 576 sin2x + 2 × 7× 24 cos x sin x

= 49 – 49 sin2 x + 576 – 576 cos2 x + 2 × 7 × 24 cos x sin x

= 625 – (7 sin x – 24 cos x)2

∴ Maximum value = 25

For maximum value

Cos x = \(\frac{-7}{25}\) and sin x =\(\frac{-24}{25}\)

∴ Minimum value = 7(\(\frac{-7}{25}\)) + 24 (\(\frac{-24}{25}\))

\(=\frac{-49-576}{25}\) = -25

∴ Minimum value = – 25

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