Solve for x sin(90 degrees -x)=-( square root of 3)/2

$\sin (90 ° - x) = - \frac{\sqrt{3}}{2}$

Step 1

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

$90 ° - x = \arcsin (- \frac{\sqrt{3}}{2})$

Step 2

Simplify the right side.

The exact value of $\arcsin (- \frac{\sqrt{3}}{2})$ is $- 60$ .

$90 ° - x = - 60$

Step 3

Move all terms not containing $x$ to the right side of the equation.

Subtract $90 °$ from both sides of the equation.

$- x = - 60 - 90 °$

Subtract $90 °$ from $- 60$ .

$- x = - 150$

Step 4

Divide each term in $- x = - 150$ by $- 1$ and simplify.

Divide each term in $- x = - 150$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 150}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 150}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 150}{- 1}$

Simplify the right side.

Divide $- 150$ by $- 1$ .

$x = 150$

Step 5

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.

$90 ° - x = 360 + 60 + 180$

Step 6

Simplify the expression to find the second solution.

Subtract $360 °$ from $360 + 60 + 180 °$ .

$90 ° - x = 360 + 60 + 180 ° - 360 °$

The resulting angle of $240 °$ is positive, less than $360 °$ , and coterminal with $360 + 60 + 180$ .

$90 ° - x = 240 °$

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Subtract $90 °$ from both sides of the equation.

$- x = 240 ° - 90 °$

Subtract $90 °$ from $240 °$ .

$- x = 150$

Divide each term in $- x = 150$ by $- 1$ and simplify.

Divide each term in $- x = 150$ by $- 1$ .

$\frac{- x}{- 1} = \frac{150}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{150}{- 1}$

Divide $x$ by $1$ .

$x = \frac{150}{- 1}$

Simplify the right side.

Divide $150$ by $- 1$ .

$x = - 150$

Step 7

Find the period of $\sin (90 ° - x)$ .

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

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

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

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

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

$\frac{360}{1}$

Divide $360$ by $1$ .

$360$

Step 8

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

Add $360$ to $- 150$ to find the positive angle.

$- 150 + 360$

Subtract $150$ from $360$ .

$210$

List the new angles.

$x = 210$

Step 9

The period of the $\sin (90 ° - x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 150 + 360 n, 210 + 360 n$ , for any integer $n$

Solve for x sin(90 degrees -x)=-( square root of 3)/2

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