Trigonometry Examples

$6 \sin (A) + 4 = 2 \sin (A) + 8$

Step 1

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

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Step 1.1

Subtract $2 \sin (A)$ from both sides of the equation.

$6 \sin (A) + 4 - 2 \sin (A) = 8$

Step 1.2

Subtract $2 \sin (A)$ from $6 \sin (A)$ .

$4 \sin (A) + 4 = 8$

Step 2

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

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Step 2.1

Subtract $4$ from both sides of the equation.

$4 \sin (A) = 8 - 4$

Step 2.2

Subtract $4$ from $8$ .

$4 \sin (A) = 4$

Step 3

Divide each term in $4 \sin (A) = 4$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 \sin (A) = 4$ by $4$ .

$\frac{4 \sin (A)}{4} = \frac{4}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 \sin (A)}{4} = \frac{4}{4}$

Step 3.2.1.2

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

$\sin (A) = \frac{4}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $4$ by $4$ .

$\sin (A) = 1$

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 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.

$A = 180 - 90$

Step 7

Subtract $90$ from $180$ .

$A = 90$

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.

$A = 90 + 360 n$ , for any integer $n$