Trigonometry Examples

$10 \sin (A) + 16 = 3 \sin (A) + 9$

Step 1

Move all terms containing $\sin (A)$ to the left side of the equation.

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Step 1.1

Subtract $3 \sin (A)$ from both sides of the equation.

$10 \sin (A) + 16 - 3 \sin (A) = 9$

Step 1.2

Subtract $3 \sin (A)$ from $10 \sin (A)$ .

$7 \sin (A) + 16 = 9$

Step 2

Move all terms not containing $\sin (A)$ to the right side of the equation.

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Step 2.1

Subtract $16$ from both sides of the equation.

$7 \sin (A) = 9 - 16$

Step 2.2

Subtract $16$ from $9$ .

$7 \sin (A) = - 7$

Step 3

Divide each term in $7 \sin (A) = - 7$ by $7$ and simplify.

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Step 3.1

Divide each term in $7 \sin (A) = - 7$ by $7$ .

$\frac{7 \sin (A)}{7} = \frac{- 7}{7}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $7$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{7 \sin (A)}{7} = \frac{- 7}{7}$

Step 3.2.1.2

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

$\sin (A) = \frac{- 7}{7}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 7$ by $7$ .

$\sin (A) = - 1$

Step 4

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

$A = \arcsin (- 1)$

Step 5

Simplify the right side.

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Step 5.1

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

$A = - 90$

Step 6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $360$ , to find a reference angle. Next, add this reference angle to $180$ to find the solution in the third quadrant.

$A = 360 + 90 + 180$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Subtract $360 °$ from $360 + 90 + 180 °$ .

$A = 360 + 90 + 180 ° - 360 °$

Step 7.2

The resulting angle of $270 °$ is positive, less than $360 °$ , and coterminal with $360 + 90 + 180$ .

$A = 270 °$

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

Add $360$ to every negative angle to get positive angles.

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Step 9.1

Add $360$ to $- 90$ to find the positive angle.

$- 90 + 360$

Step 9.2

Subtract $90$ from $360$ .

$270$

Step 9.3

List the new angles.

$A = 270$

Step 10

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

$A = 270 + 360 n, 270 + 360 n$ , for any integer $n$

Step 11

Consolidate the answers.

$A = 270 + 360 n$ , for any integer $n$