Reduce 24 cos x + 7 sin x to the Sine and Cosine of a Single Expression

Solution

We use the standard form:

\[ a\cos x+b\sin x=R\cos(x-\alpha) \]

where

\[ R=\sqrt{a^2+b^2} \]

Given expression:

\[ 24\cos x+7\sin x \]

Here,

\[ a=24, \qquad b=7 \]

Now,

\[ R=\sqrt{24^2+7^2} \]

\[ =\sqrt{576+49} \]

\[ =\sqrt{625} \]

\[ =25 \]

Let

\[ 24\cos x+7\sin x = 25\cos(x-\alpha) \]

Using the identity:

\[ 25\cos(x-\alpha) = 25(\cos x\cos\alpha+\sin x\sin\alpha) \]

Comparing coefficients,

\[ 25\cos\alpha=24 \]

\[ \cos\alpha=\frac{24}{25} \]

and

\[ 25\sin\alpha=7 \]

\[ \sin\alpha=\frac{7}{25} \]

Therefore,

\[ \alpha=\tan^{-1}\left(\frac{7}{24}\right) \]

Hence,

\[ \boxed{ 24\cos x+7\sin x = 25\cos\left(x-\tan^{-1}\frac{7}{24}\right) } \]

Also, using sine form:

\[ \boxed{ 24\cos x+7\sin x = 25\sin\left(x+\tan^{-1}\frac{24}{7}\right) } \]

Final Answer

\[ \boxed{ 24\cos x+7\sin x = 25\cos\left(x-\tan^{-1}\frac{7}{24}\right) } \]

or

\[ \boxed{ 24\cos x+7\sin x = 25\sin\left(x+\tan^{-1}\frac{24}{7}\right) } \]