Trigonometry Examples

$\sin (x + 30) = \cos (20)$

Step 1

Evaluate $\cos (20)$ .

$\sin (x + 30) = 0.93969262$

Step 2

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

$x + 30 = \arcsin (0.93969262)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (0.93969262)$ .

$x + 30 = 70$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Subtract $30$ from both sides of the equation.

$x = 70 - 30$

Step 4.2

Subtract $30$ from $70$ .

$x = 40$

Step 5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$x + 30 = 180 - 70$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $70$ from $180$ .

$x + 30 = 110$

Step 6.2

Move all terms not containing $x$ to the right side of the equation.

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Step 6.2.1

Subtract $30$ from both sides of the equation.

$x = 110 - 30$

Step 6.2.2

Subtract $30$ from $110$ .

$x = 80$

Step 7

Find the period of $\sin (x + 30)$ .

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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 $\sin (x + 30)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 40 + 360 n, 80 + 360 n$ , for any integer $n$