Trigonometry Examples

$2 \sin (2 (x + \frac{π}{6})) + 5 = 2 \cos (2 (x - \frac{π}{6})) + 5$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$2 \sin (2 x + 2 \frac{π}{6}) + 5 = 2 \cos (2 (x - \frac{π}{6})) + 5$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Factor $2$ out of $6$ .

$2 \sin (2 x + 2 \frac{π}{2 (3)}) + 5 = 2 \cos (2 (x - \frac{π}{6})) + 5$

Step 1.2.2

Cancel the common factor.

$2 \sin (2 x + 2 \frac{π}{2 \cdot 3}) + 5 = 2 \cos (2 (x - \frac{π}{6})) + 5$

Step 1.2.3

Rewrite the expression.

$2 \sin (2 x + \frac{π}{3}) + 5 = 2 \cos (2 (x - \frac{π}{6})) + 5$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$2 \sin (2 x + \frac{π}{3}) + 5 = 2 \cos (2 x + 2 (- \frac{π}{6})) + 5$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$2 \sin (2 x + \frac{π}{3}) + 5 = 2 \cos (2 x + 2 \frac{- π}{6}) + 5$

Step 2.2.2

Factor $2$ out of $6$ .

$2 \sin (2 x + \frac{π}{3}) + 5 = 2 \cos (2 x + 2 \frac{- π}{2 (3)}) + 5$

Step 2.2.3

Cancel the common factor.

$2 \sin (2 x + \frac{π}{3}) + 5 = 2 \cos (2 x + 2 \frac{- π}{2 \cdot 3}) + 5$

Step 2.2.4

Rewrite the expression.

$2 \sin (2 x + \frac{π}{3}) + 5 = 2 \cos (2 x + \frac{- π}{3}) + 5$

Step 2.3

Move the negative in front of the fraction.

$2 \sin (2 x + \frac{π}{3}) + 5 = 2 \cos (2 x - \frac{π}{3}) + 5$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{π}{8} + \frac{π n}{2}$ , for any integer $n$

Step 4