Trigonometry Examples

$3 \sin (x) + 4 \cos (x)$

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 $3 \sin (x)$ and $4 \cos (x)$ into $k = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise $3$ to the power of $2$ .

$k = \sqrt{9 + {(4)}^{2}}$

Step 2.2

Raise $4$ to the power of $2$ .

$k = \sqrt{9 + 16}$

Step 2.3

Add $9$ and $16$ .

$k = \sqrt{25}$

Step 2.4

Rewrite $25$ as $5^{2}$ .

$k = \sqrt{5^{2}}$

Step 2.5

Pull terms out from under the radical, assuming positive real numbers.

$k = 5$

Step 3

Find the value for $θ$ by substituting the coefficients from $3 \sin (x)$ and $4 \cos (x)$ into $θ = \tan^{-1} (\frac{b}{a})$ .

$θ = \tan^{-1} (\frac{4}{3})$

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 $θ$ .

$5 \sin ((x + θ = 0.92729521))$