Trigonometry Examples

$\cos (20) + 4 \sin^{2} (x) = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\cos (20) + 4 \sin^{2} (x) - 2 = 0$

Step 2

Simplify $\cos (20) + 4 \sin^{2} (x) - 2$ .

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Step 2.1

Evaluate $\cos (20)$ .

$0.93969262 + 4 \sin^{2} (x) - 2 = 0$

Step 2.2

Subtract $2$ from $0.93969262$ .

$- 1.06030737 + 4 \sin^{2} (x) = 0$

Step 3

Solve for $x$ .

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Step 3.1

Add $1.06030737$ to both sides of the equation.

$4 \sin^{2} (x) = 1.06030737$

Step 3.2

Divide each term in $4 \sin^{2} (x) = 1.06030737$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 \sin^{2} (x) = 1.06030737$ by $4$ .

$\frac{4 \sin^{2} (x)}{4} = \frac{1.06030737}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 \sin^{2} (x)}{4} = \frac{1.06030737}{4}$

Step 3.2.2.1.2

Divide $\sin^{2} (x)$ by $1$ .

$\sin^{2} (x) = \frac{1.06030737}{4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $1.06030737$ by $4$ .

$\sin^{2} (x) = 0.26507684$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sin (x) = \pm \sqrt{0.26507684}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$\sin (x) = \sqrt{0.26507684}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sin (x) = - \sqrt{0.26507684}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sin (x) = \sqrt{0.26507684}, - \sqrt{0.26507684}$

Step 3.5

Set up each of the solutions to solve for $x$ .

$\sin (x) = \sqrt{0.26507684}$

$\sin (x) = - \sqrt{0.26507684}$

Step 3.6

Solve for $x$ in $\sin (x) = \sqrt{0.26507684}$ .

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Step 3.6.1

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

$x = \arcsin (\sqrt{0.26507684})$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Evaluate $\arcsin (\sqrt{0.26507684})$ .

$x = 30.98783966$

Step 3.6.3

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.

$x = 180 - 30.98783966$

Step 3.6.4

Subtract $30.98783966$ from $180$ .

$x = 149.01216033$

Step 3.6.5

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

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Step 3.6.5.1

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

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

Step 3.6.5.2

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

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

Step 3.6.5.3

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

$\frac{360}{1}$

Step 3.6.5.4

Divide $360$ by $1$ .

$360$

Step 3.6.6

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

$x = 30.98783966 + 360 n, 149.01216033 + 360 n$ , for any integer $n$

Step 3.7

Solve for $x$ in $\sin (x) = - \sqrt{0.26507684}$ .

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Step 3.7.1

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

$x = \arcsin (- \sqrt{0.26507684})$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Evaluate $\arcsin (- \sqrt{0.26507684})$ .

$x = - 30.98783966$

Step 3.7.3

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.

$x = 360 + 30.98783966 + 180$

Step 3.7.4

Simplify the expression to find the second solution.

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Step 3.7.4.1

Subtract $360 °$ from $360 + 30.98783966 + 180 °$ .

$x = 360 + 30.98783966 + 180 ° - 360 °$

Step 3.7.4.2

The resulting angle of $210.98783966 °$ is positive, less than $360 °$ , and coterminal with $360 + 30.98783966 + 180$ .

$x = 210.98783966 °$

Step 3.7.5

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

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Step 3.7.5.1

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

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

Step 3.7.5.2

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

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

Step 3.7.5.3

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

$\frac{360}{1}$

Step 3.7.5.4

Divide $360$ by $1$ .

$360$

Step 3.7.6

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

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Step 3.7.6.1

Add $360$ to $- 30.98783966$ to find the positive angle.

$- 30.98783966 + 360$

Step 3.7.6.2

Subtract $30.98783966$ from $360$ .

$329.01216033$

Step 3.7.6.3

List the new angles.

$x = 329.01216033$

Step 3.7.7

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

$x = 210.98783966 + 360 n, 329.01216033 + 360 n$ , for any integer $n$

Step 3.8

List all of the solutions.

$x = 30.98783966 + 360 n, 149.01216033 + 360 n, 210.98783966 + 360 n, 329.01216033 + 360 n$ , for any integer $n$

Step 3.9

Consolidate the solutions.

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Step 3.9.1

Consolidate $30.98783966 + 360 n$ and $210.98783966 + 360 n$ to $30.98783966 + 180 n$ .

$x = 30.98783966 + 180 n, 149.01216033 + 360 n, 329.01216033 + 360 n$ , for any integer $n$

Step 3.9.2

Consolidate $149.01216033 + 360 n$ and $329.01216033 + 360 n$ to $149.01216033 + 180 n$ .

$x = 30.98783966 + 180 n, 149.01216033 + 180 n$ , for any integer $n$