Precalculus Examples

$\cos (2 x) + 2 \sin^{2} (x)$

Step 1

Given the expression $a \sin (x) + b \cos (x)$ , find the values of $k$ and $θ$ .

$k = \sqrt{a^{2} + b^{2}}$

$θ = \tan^{-1} (\frac{b}{a})$

Step 2

Calculate the value for $k$ by substituting the coefficients from $2 \sin^{2} (x)$ and $\cos (2 x)$ into $k = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise $2$ to the power of $2$ .

$k = \sqrt{4 + {(1)}^{2}}$

Step 2.2

One to any power is one.

$k = \sqrt{4 + 1}$

Step 2.3

Add $4$ and $1$ .

$k = \sqrt{5}$

Step 3

Find the value for $θ$ by substituting the coefficients from $2 \sin^{2} (x)$ and $\cos (2 x)$ into $θ = \tan^{-1} (\frac{b}{a})$ .

$θ = \tan^{-1} (\frac{1}{2})$

Step 4

Linear combinations of trigonometric functions dictate that $a \sin (x) + b \cos (x) = k \sin (x + θ)$ . Substitute the values of $k$ and $θ$ .

$\sqrt{5} \sin ((x + θ = 0.4636476))$