Trigonometry Examples

$\sin (2 x) + \cos (x)$

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

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

$θ = \tan^{-1} (\frac{b}{a})$

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

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One to any power is one.

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

One to any power is one.

$k = \sqrt{1 + 1}$

Add $1$ and $1$ .

$k = \sqrt{2}$

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

$θ = \tan^{-1} (\frac{1}{1})$

Divide $1$ by $1$ .

$\tan^{-1} (1)$

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

$\sqrt{2} \sin (x + \frac{π}{4})$