Trigonometry Examples

$\cos (2 x) = \sin (1)$

Step 1

Evaluate $\sin (1)$ .

$\cos (2 x) = 0.0174524$

Step 2

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

$2 x = \arccos (0.0174524)$

Step 3

Simplify the right side.

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Evaluate $\arccos (0.0174524)$ .

$2 x = 89$

Step 4

Divide each term in $2 x = 89$ by $2$ and simplify.

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Divide each term in $2 x = 89$ by $2$ .

$\frac{2 x}{2} = \frac{89}{2}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{89}{2}$

Divide $x$ by $1$ .

$x = \frac{89}{2}$

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.

$2 x = 360 - 89$

Step 6

Solve for $x$ .

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Subtract $89$ from $360$ .

$2 x = 271$

Divide each term in $2 x = 271$ by $2$ and simplify.

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Divide each term in $2 x = 271$ by $2$ .

$\frac{2 x}{2} = \frac{271}{2}$

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{271}{2}$

Divide $x$ by $1$ .

$x = \frac{271}{2}$

Step 7

Find the period of $\cos (2 x)$ .

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

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

Replace $b$ with $2$ in the formula for period.

$\frac{360}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{360}{2}$

Divide $360$ by $2$ .

$180$

Step 8

The period of the $\cos (2 x)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$x = \frac{89}{2} + 180 n, \frac{271}{2} + 180 n$ , for any integer $n$