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Question
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
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Solution
\[\text{ Let } f(x) = 24 \cos x + 7\sin x\]
\[\text{ Dividing and multiplying by } \sqrt{{24}^2 + 7^2}, i . e . \text{ by 25, we get }: \]
\[ f(x) = 25\left( \frac{24}{25} \cos x + \frac{7}{25}\sin x \right)\]
\[ \Rightarrow f(x) = 25(\sin\alpha \cos x + \cos\alpha \sin x), \text{ where } \sin\alpha = \frac{24}{25} and \cos\alpha = \frac{7}{25}\]
\[ \Rightarrow f(x) = 25 \sin(\alpha + x), \text{ where } \tan\alpha = \frac{24}{7} . \]
\[\text{ Again }, \]
\[ f(x) = 25\left( \frac{24}{25} \cos x + \frac{7}{25}\sin x \right)\]
\[ \Rightarrow f(x) = 25(\cos\alpha \cos x + \sin\alpha \sin x), \text{ where } \cos\alpha = \frac{24}{25}, \sin\alpha = \frac{7}{25} . \]
\[ \Rightarrow f(x) = 25 \cos(\alpha - x), \text{ where }\tan\alpha = \frac{7}{24} .\]
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