Trigonometry Examples

$\sin (60) = \cos (x)$

Step 1

Rewrite the equation as $\cos (x) = \sin (60)$ .

$\cos (x) = \sin (60)$

Step 2

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\cos (x) = \frac{\sqrt{3}}{2}$

Step 3

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (\frac{\sqrt{3}}{2})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $30$ .

$x = 30$

Step 5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the fourth quadrant.

$x = 360 - 30$

Step 6

Subtract $30$ from $360$ .

$x = 330$

Step 7

Find the period of $\cos (x)$ .

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Step 7.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 7.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 7.4

Divide $360$ by $1$ .

$360$

Step 8

The period of the $\cos (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 30 + 360 n, 330 + 360 n$ , for any integer $n$