The expression you've written, ( 28282993\sin(x) + 16\cos(x) ), represents a linear combination of sine and cosine functions. To simplify or rewrite this expression, you can use the formula for representing it as a single sinusoidal function:
[
R\sin(x + \phi) = a\sin(x) + b\cos(x)
]
where ( a = 28282993 ) and ( b = 16 ), and ( R ) and ( \phi ) are given by:
[
R = \sqrt{a^2 + b^2}
]
[
\phi = \tan^{-1}\left(\frac{b}{a}\right)
]
Step 1: Calculate ( R )
[
R = \sqrt{(28282993)^2 + (16)^2}
]
[
R = \sqrt{28282993^2 + 256}
]
[
= \sqrt{799,588,806,954,864 + 256} \approx 28282993 \quad (\text{since } 256 \text{ is negligible})
]
Step 2: Calculate ( \phi )
[
\phi = \tan^{-1}\left(\frac{16}{28282993}\right)
]
Since ( \frac{16}{28282993} ) is a very small number, ( \phi ) will also be a very small angle.
Final Representation
Thus, the expression can be rewritten as:
[
28282993\sin(x) + 16\cos(x) \approx 28282993\sin(x + \phi)
]
You may further evaluate ( \phi ) if needed, or you can leave the expression in this form indicating that it represents a single sinusoidal function.
