Trigonometry Examples

$\sin (x) = \frac{62 \sin (47)}{70}$

Step 1

Simplify $\frac{62 \sin (47)}{70}$ .

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Step 1.1

Cancel the common factor of $62$ and $70$ .

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Step 1.1.1

Factor $2$ out of $62 \sin (47)$ .

$\sin (x) = \frac{2 (31 \sin (47))}{70}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $70$ .

$\sin (x) = \frac{2 (31 \sin (47))}{2 (35)}$

Step 1.1.2.2

Cancel the common factor.

$\sin (x) = \frac{2 (31 \sin (47))}{2 \cdot 35}$

Step 1.1.2.3

Rewrite the expression.

$\sin (x) = \frac{31 \sin (47)}{35}$

Step 1.2

Evaluate $\sin (47)$ .

$\sin (x) = \frac{31 \cdot 0.7313537}{35}$

Step 1.3

Multiply $31$ by $0.7313537$ .

$\sin (x) = \frac{22.67196475}{35}$

Step 1.4

Divide $22.67196475$ by $35$ .

$\sin (x) = 0.64777042$

Step 2

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

$x = \arcsin (0.64777042)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (0.64777042)$ .

$x = 40.37371136$

Step 4

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.

$x = 180 - 40.37371136$

Step 5

Subtract $40.37371136$ from $180$ .

$x = 139.62628863$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

$\frac{360}{1}$

Step 6.4

Divide $360$ by $1$ .

$360$

Step 7

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

$x = 40.37371136 + 360 n, 139.62628863 + 360 n$ , for any integer $n$