Precalculus Examples

$\sin (\frac{π}{12}) \cos (\frac{7 π}{12}) + \cos (\frac{π}{12}) \sin (\frac{7 π}{12})$

Step 1

Given the expression $a \sin (π) + b \cos (π)$ , 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 $\sin (\frac{π}{12}) \cos (\frac{7 π}{12})$ and $\cos (\frac{π}{12}) \sin (\frac{7 π}{12})$ into $k = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

One to any power is one.

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

Step 2.2

One to any power is one.

$k = \sqrt{1 + 1}$

Step 2.3

Add $1$ and $1$ .

$k = \sqrt{2}$

Step 3

Find the value for $θ$ by substituting the coefficients from $\sin (\frac{π}{12}) \cos (\frac{7 π}{12})$ and $\cos (\frac{π}{12}) \sin (\frac{7 π}{12})$ into $θ = \tan^{-1} (\frac{b}{a})$ .

$θ = \tan^{-1} (\frac{1}{1})$

Step 4

Divide $1$ by $1$ .

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

Step 5

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

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