Trigonometry Examples

$4.1042 \sin (90) = 22 \sin (A)$

Step 1

Rewrite the equation as $22 \sin (A) = 4.1042 \sin (90)$ .

$22 \sin (A) = 4.1042 \sin (90)$

Step 2

Simplify.

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Step 2.1

The exact value of $\sin (90)$ is $1$ .

$22 \sin (A) = 4.1042 \cdot 1$

Step 2.2

Multiply $4.1042$ by $1$ .

$22 \sin (A) = 4.1042$

Step 3

Divide each term in $22 \sin (A) = 4.1042$ by $22$ and simplify.

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Step 3.1

Divide each term in $22 \sin (A) = 4.1042$ by $22$ .

$\frac{22 \sin (A)}{22} = \frac{4.1042}{22}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $22$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{22 \sin (A)}{22} = \frac{4.1042}{22}$

Step 3.2.1.2

Divide $\sin (A)$ by $1$ .

$\sin (A) = \frac{4.1042}{22}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $4.1042$ by $22$ .

$\sin (A) = 0.1865\overline{54}$

Step 4

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

$A = \arcsin (0.1865\overline{54})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (0.1865\overline{54})$ .

$A = 10.75177929$

Step 6

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.

$A = 180 - 10.75177929$

Step 7

Subtract $10.75177929$ from $180$ .

$A = 169.2482207$

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

$\frac{360}{1}$

Step 8.4

Divide $360$ by $1$ .

$360$

Step 9

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

$A = 10.75177929 + 360 n, 169.2482207 + 360 n$ , for any integer $n$