Trigonometry Examples

$\csc (\frac{3 π}{4}) = \sin (x)$

Step 1

Rewrite the equation as $\sin (x) = \csc (\frac{3 π}{4})$ .

$\sin (x) = \csc (\frac{3 π}{4})$

Step 2

Simplify the expression using the $h a l f - a n g l e$ formula.

$\pm \sqrt{1 - \frac{\cos (2 x)}{2}} = \csc (\frac{3 π}{4})$

Step 3

Remove the $\pm$ from both sides of the equation.

$\sqrt{1 - \frac{\cos (2 x)}{2}} = \csc (\frac{3 π}{4})$

Step 4

Since the square roots of each expression are equal, the expression inside the square root must also be equal.

$1 - \frac{\cos (2 x)}{2} = \csc (\frac{3 π}{4})$

Step 5

Simplify $\csc (\frac{3 π}{4})$ .

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Step 5.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$1 - \frac{\cos (2 x)}{2} = \csc (\frac{π}{4})$

Step 5.2

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

$1 - \frac{\cos (2 x)}{2} = \sqrt{2}$

Step 6

Subtract $1$ from both sides of the equation.

$- \frac{\cos (2 x)}{2} = \sqrt{2} - 1$

Step 7

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{\cos (2 x)}{2}) = - 2 (\sqrt{2} - 1)$

Step 8

Simplify both sides of the equation.

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Step 8.1

Simplify the left side.

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Step 8.1.1

Simplify $- 2 (- \frac{\cos (2 x)}{2})$ .

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Step 8.1.1.1

Cancel the common factor of $2$ .

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Step 8.1.1.1.1

Move the leading negative in $- \frac{\cos (2 x)}{2}$ into the numerator.

$- 2 \frac{- \cos (2 x)}{2} = - 2 (\sqrt{2} - 1)$

Step 8.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- \cos (2 x)}{2} = - 2 (\sqrt{2} - 1)$

Step 8.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \cos (2 x)}{2} = - 2 (\sqrt{2} - 1)$

Step 8.1.1.1.4

Rewrite the expression.

$- - \cos (2 x) = - 2 (\sqrt{2} - 1)$

Step 8.1.1.2

Multiply.

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Step 8.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 \cos (2 x) = - 2 (\sqrt{2} - 1)$

Step 8.1.1.2.2

Multiply $\cos (2 x)$ by $1$ .

$\cos (2 x) = - 2 (\sqrt{2} - 1)$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $- 2 (\sqrt{2} - 1)$ .

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Step 8.2.1.1

Apply the distributive property.

$\cos (2 x) = - 2 \sqrt{2} - 2 \cdot - 1$

Step 8.2.1.2

Multiply $- 2$ by $- 1$ .

$\cos (2 x) = - 2 \sqrt{2} + 2$

Step 9

Convert the right side of the equation to its decimal equivalent.

$\cos (2 x) = - 0.82842712$

Step 10

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

$2 x = \arccos (- 0.82842712)$

Step 11

Simplify the right side.

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Step 11.1

Evaluate $\arccos (- 0.82842712)$ .

$2 x = 2.54708993$

Step 12

Divide each term in $2 x = 2.54708993$ by $2$ and simplify.

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Step 12.1

Divide each term in $2 x = 2.54708993$ by $2$ .

$\frac{2 x}{2} = \frac{2.54708993}{2}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $2$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2.54708993}{2}$

Step 12.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.54708993}{2}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Divide $2.54708993$ by $2$ .

$x = 1.27354496$

Step 13

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$2 x = 2 (3.14159265) - 2.54708993$

Step 14

Solve for $x$ .

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Step 14.1

Simplify.

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Step 14.1.1

Multiply $2$ by $3.14159265$ .

$2 x = 6.2831853 - 2.54708993$

Step 14.1.2

Subtract $2.54708993$ from $6.2831853$ .

$2 x = 3.73609537$

Step 14.2

Divide each term in $2 x = 3.73609537$ by $2$ and simplify.

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Step 14.2.1

Divide each term in $2 x = 3.73609537$ by $2$ .

$\frac{2 x}{2} = \frac{3.73609537}{2}$

Step 14.2.2

Simplify the left side.

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Step 14.2.2.1

Cancel the common factor of $2$ .

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Step 14.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3.73609537}{2}$

Step 14.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3.73609537}{2}$

Step 14.2.3

Simplify the right side.

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Step 14.2.3.1

Divide $3.73609537$ by $2$ .

$x = 1.86804768$

Step 15

Find the period of $\cos (2 x)$ .

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Step 15.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 15.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 15.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 15.4.2

Divide $π$ by $1$ .

$π$

Step 16

The period of the $\cos (2 x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 1.27354496 + π n, 1.86804768 + π n$ , for any integer $n$

Step 17

Exclude the solutions that do not make $\csc (\frac{3 π}{4}) = \sin (x)$ true.

No solution