Trigonometry Examples

\sin (x) + 1 = 0

$\sin (x) + 1 = 0$

Step 1

Subtract

1

$1$ from both sides of the equation.

\sin (x) = - 1

$\sin (x) = - 1$

Step 2

Take the inverse sine of both sides of the equation to extract

x

$x$ from inside the sine.

x = \arcsin (- 1)

$x = \arcsin (- 1)$

Step 3

Simplify the right side.

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Step 3.1

The exact value of

\arcsin (- 1)

$\arcsin (- 1)$ is

- 90

$- 90$ .

x = - 90

$x = - 90$

x = - 90

$x = - 90$

Step 4

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from

360

$360$ , to find a reference angle. Next, add this reference angle to

180

$180$ to find the solution in the third quadrant.

x = 360 + 90 + 180

$x = 360 + 90 + 180$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Subtract

360 °

$360 °$ from

360 + 90 + 180 °

$360 + 90 + 180 °$ .

x = 360 + 90 + 180 ° - 360 °

$x = 360 + 90 + 180 ° - 360 °$

Step 5.2

The resulting angle of

270 °

$270 °$ is positive, less than

360 °

$360 °$ , and coterminal with

360 + 90 + 180

$360 + 90 + 180$ .

x = 270 °

$x = 270 °$

x = 270 °

$x = 270 °$

Step 6

Find the period of

\sin (x)

$\sin (x)$ .

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Step 6.1

The period of the function can be calculated using

\frac{360}{| b |}

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

\frac{360}{| b |}

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

Step 6.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{360}{| 1 |}

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

Step 6.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{360}{1}

$\frac{360}{1}$

Step 6.4

Divide

360

$360$ by

1

$1$ .

360

$360$

360

$360$

Step 7

Add

360

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

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Step 7.1

Add

360

$360$ to

- 90

$- 90$ to find the positive angle.

- 90 + 360

$- 90 + 360$

Step 7.2

Subtract

90

$90$ from

360

$360$ .

270

$270$

Step 7.3

List the new angles.

x = 270

$x = 270$

x = 270

$x = 270$

Step 8

The period of the

\sin (x)

$\sin (x)$ function is

360

$360$ so values will repeat every

360

$360$ degrees in both directions.

x = 270 + 360 n, 270 + 360 n

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

n

$n$

Step 9

Consolidate the answers.

x = 270 + 360 n

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

n

$n$