Trigonometry Examples

$\frac{{(\sin (x) + \cos (x))}^{2}}{1 + 2 \sin (x) \cos (x)}$

Step 1

Interchange the variables.

$x = \frac{{(\sin (y) + \cos (y))}^{2}}{1 + 2 \sin (y) \cos (y)}$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $\frac{{(\sin (y) + \cos (y))}^{2}}{1 + 2 \sin (y) \cos (y)} = x$ .

$\frac{{(\sin (y) + \cos (y))}^{2}}{1 + 2 \sin (y) \cos (y)} = x$

Step 2.2

Divide each term in the equation by $\cos (y)$ .

$\frac{\frac{{(\sin (y) + \cos (y))}^{2}}{1 + 2 \sin (y) \cos (y)}}{\cos (y)} = \frac{x}{\cos (y)}$

Step 2.3

Separate fractions.

$\frac{\frac{{(\sin (y) + \cos (y))}^{2}}{1 + 2 \sin (y) \cos (y)}}{1} \cdot \frac{1}{\cos (y)} = \frac{x}{\cos (y)}$

Step 2.4

Convert from $\frac{1}{\cos (y)}$ to $\sec (y)$ .

$\frac{\frac{{(\sin (y) + \cos (y))}^{2}}{1 + 2 \sin (y) \cos (y)}}{1} \cdot \sec (y) = \frac{x}{\cos (y)}$

Step 2.5

Divide $\frac{{(\sin (y) + \cos (y))}^{2}}{1 + 2 \sin (y) \cos (y)}$ by $1$ .

$\frac{{(\sin (y) + \cos (y))}^{2}}{1 + 2 \sin (y) \cos (y)} \cdot \sec (y) = \frac{x}{\cos (y)}$

Step 2.6

Apply the sine double-angle identity.

$\frac{{(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} \cdot \sec (y) = \frac{x}{\cos (y)}$

Step 2.7

Combine fractions.

Tap for more steps...

Step 2.7.1

Combine $\frac{{(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)}$ and $\sec (y)$ .

$\frac{{(\sin (y) + \cos (y))}^{2} \sec (y)}{1 + \sin (2 y)} = \frac{x}{\cos (y)}$

Step 2.7.2

Reorder factors in $\frac{{(\sin (y) + \cos (y))}^{2} \sec (y)}{1 + \sin (2 y)}$ .

$\frac{\sec (y) {(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} = \frac{x}{\cos (y)}$

Step 2.8

Separate fractions.

$\frac{\sec (y) {(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} = \frac{x}{1} \cdot \frac{1}{\cos (y)}$

Step 2.9

Convert from $\frac{1}{\cos (y)}$ to $\sec (y)$ .

$\frac{\sec (y) {(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} = \frac{x}{1} \cdot \sec (y)$

Step 2.10

Divide $x$ by $1$ .

$\frac{\sec (y) {(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} = x \sec (y)$

Step 2.11

Simplify the left side.

Tap for more steps...

Step 2.11.1

Simplify $\frac{\sec (y) {(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)}$ .

Tap for more steps...

Step 2.11.1.1

Rewrite $\sec (y)$ in terms of sines and cosines.

$\frac{\frac{1}{\cos (y)} {(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} = x \sec (y)$

Step 2.11.1.2

Combine $\frac{1}{\cos (y)}$ and ${(\sin (y) + \cos (y))}^{2}$ .

$\frac{\frac{{(\sin (y) + \cos (y))}^{2}}{\cos (y)}}{1 + \sin (2 y)} = x \sec (y)$

Step 2.11.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{{(\sin (y) + \cos (y))}^{2}}{\cos (y)} \cdot \frac{1}{1 + \sin (2 y)} = x \sec (y)$

Step 2.11.1.4

Multiply $\frac{{(\sin (y) + \cos (y))}^{2}}{\cos (y)}$ by $\frac{1}{1 + \sin (2 y)}$ .

$\frac{{(\sin (y) + \cos (y))}^{2}}{\cos (y) (1 + \sin (2 y))} = x \sec (y)$

Step 2.12

Simplify the right side.

Tap for more steps...

Step 2.12.1

Simplify $x \sec (y)$ .

Tap for more steps...

Step 2.12.1.1

Rewrite $\sec (y)$ in terms of sines and cosines.

$\frac{{(\sin (y) + \cos (y))}^{2}}{\cos (y) (1 + \sin (2 y))} = x \frac{1}{\cos (y)}$

Step 2.12.1.2

Combine $x$ and $\frac{1}{\cos (y)}$ .

$\frac{{(\sin (y) + \cos (y))}^{2}}{\cos (y) (1 + \sin (2 y))} = \frac{x}{\cos (y)}$

Step 2.13

Multiply both sides of the equation by $\cos (y)$ .

$\cos (y) \frac{{(\sin (y) + \cos (y))}^{2}}{\cos (y) (1 + \sin (2 y))} = \cos (y) \frac{x}{\cos (y)}$

Step 2.14

Cancel the common factor of $\cos (y)$ .

Tap for more steps...

Step 2.14.1

Cancel the common factor.

$\cos (y) \frac{{(\sin (y) + \cos (y))}^{2}}{\cos (y) (1 + \sin (2 y))} = \cos (y) \frac{x}{\cos (y)}$

Step 2.14.2

Rewrite the expression.

$\frac{{(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} = \cos (y) \frac{x}{\cos (y)}$

Step 2.15

Cancel the common factor of $\cos (y)$ .

Tap for more steps...

Step 2.15.1

Cancel the common factor.

$\frac{{(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} = \cos (y) \frac{x}{\cos (y)}$

Step 2.15.2

Rewrite the expression.

$\frac{{(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} = x$

Step 2.16

Multiply both sides by $1 + \sin (2 y)$ .

$\frac{{(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} (1 + \sin (2 y)) = x (1 + \sin (2 y))$

Step 2.17

Simplify.

Tap for more steps...

Step 2.17.1

Simplify the left side.

Tap for more steps...

Step 2.17.1.1

Cancel the common factor of $1 + \sin (2 y)$ .

Tap for more steps...

Step 2.17.1.1.1

Cancel the common factor.

$\frac{{(\sin (y) + \cos (y))}^{2}}{1 + \sin (2 y)} (1 + \sin (2 y)) = x (1 + \sin (2 y))$

Step 2.17.1.1.2

Rewrite the expression.

${(\sin (y) + \cos (y))}^{2} = x (1 + \sin (2 y))$

Step 2.17.2

Simplify the right side.

Tap for more steps...

Step 2.17.2.1

Simplify $x (1 + \sin (2 y))$ .

Tap for more steps...

Step 2.17.2.1.1

Apply the distributive property.

${(\sin (y) + \cos (y))}^{2} = x \cdot 1 + x \sin (2 y)$

Step 2.17.2.1.2

Multiply $x$ by $1$ .

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

Step 2.18

Solve for $y$ .

Tap for more steps...

Step 2.18.1

Simplify the left side.

Tap for more steps...

Step 2.18.1.1

Simplify ${(\sin (y) + \cos (y))}^{2}$ .

Tap for more steps...

Step 2.18.1.1.1

Rewrite.

$0 + 0 + {(\sin (y) + \cos (y))}^{2} = x + x \sin (2 y)$

Step 2.18.1.1.2

Rewrite ${(\sin (y) + \cos (y))}^{2}$ as $(\sin (y) + \cos (y)) (\sin (y) + \cos (y))$ .

$(\sin (y) + \cos (y)) (\sin (y) + \cos (y)) = x + x \sin (2 y)$

Step 2.18.1.1.3

Expand $(\sin (y) + \cos (y)) (\sin (y) + \cos (y))$ using the FOIL Method.

Tap for more steps...

Step 2.18.1.1.3.1

Apply the distributive property.

$\sin (y) (\sin (y) + \cos (y)) + \cos (y) (\sin (y) + \cos (y)) = x + x \sin (2 y)$

Step 2.18.1.1.3.2

Apply the distributive property.

$\sin (y) \sin (y) + \sin (y) \cos (y) + \cos (y) (\sin (y) + \cos (y)) = x + x \sin (2 y)$

Step 2.18.1.1.3.3

Apply the distributive property.

$\sin (y) \sin (y) + \sin (y) \cos (y) + \cos (y) \sin (y) + \cos (y) \cos (y) = x + x \sin (2 y)$

Step 2.18.1.1.4

Simplify and combine like terms.

Tap for more steps...

Step 2.18.1.1.4.1

Simplify each term.

Tap for more steps...

Step 2.18.1.1.4.1.1

Multiply $\sin (y) \sin (y)$ .

Tap for more steps...

Step 2.18.1.1.4.1.1.1

Raise $\sin (y)$ to the power of $1$ .

$\sin^{1} (y) \sin (y) + \sin (y) \cos (y) + \cos (y) \sin (y) + \cos (y) \cos (y) = x + x \sin (2 y)$

Step 2.18.1.1.4.1.1.2

Raise $\sin (y)$ to the power of $1$ .

$\sin^{1} (y) \sin^{1} (y) + \sin (y) \cos (y) + \cos (y) \sin (y) + \cos (y) \cos (y) = x + x \sin (2 y)$

Step 2.18.1.1.4.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\sin (y)}^{1 + 1} + \sin (y) \cos (y) + \cos (y) \sin (y) + \cos (y) \cos (y) = x + x \sin (2 y)$

Step 2.18.1.1.4.1.1.4

Add $1$ and $1$ .

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

Step 2.18.1.1.4.1.2

Multiply $\cos (y) \cos (y)$ .

Tap for more steps...

Step 2.18.1.1.4.1.2.1

Raise $\cos (y)$ to the power of $1$ .

$\sin^{2} (y) + \sin (y) \cos (y) + \cos (y) \sin (y) + \cos^{1} (y) \cos (y) = x + x \sin (2 y)$

Step 2.18.1.1.4.1.2.2

Raise $\cos (y)$ to the power of $1$ .

$\sin^{2} (y) + \sin (y) \cos (y) + \cos (y) \sin (y) + \cos^{1} (y) \cos^{1} (y) = x + x \sin (2 y)$

Step 2.18.1.1.4.1.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin^{2} (y) + \sin (y) \cos (y) + \cos (y) \sin (y) + {\cos (y)}^{1 + 1} = x + x \sin (2 y)$

Step 2.18.1.1.4.1.2.4

Add $1$ and $1$ .

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

Step 2.18.1.1.4.2

Reorder the factors of $\sin (y) \cos (y)$ .

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

Step 2.18.1.1.4.3

Add $\cos (y) \sin (y)$ and $\cos (y) \sin (y)$ .

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

Step 2.18.1.1.5

Move $\cos^{2} (y)$ .

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

Step 2.18.1.1.6

Apply pythagorean identity.

$1 + 2 \cos (y) \sin (y) = x + x \sin (2 y)$

Step 2.18.1.1.7

Simplify each term.

Tap for more steps...

Step 2.18.1.1.7.1

Reorder $2 \cos (y)$ and $\sin (y)$ .

$1 + \sin (y) (2 \cos (y)) = x + x \sin (2 y)$

Step 2.18.1.1.7.2

Reorder $\sin (y)$ and $2$ .

$1 + 2 \cdot \sin (y) \cos (y) = x + x \sin (2 y)$

Step 2.18.1.1.7.3

Apply the sine double-angle identity.

$1 + \sin (2 y) = x + x \sin (2 y)$

Step 2.18.2

Substitute $u$ for $\sin (2 y)$ .

$1 + u = x + x (u)$

Step 2.18.3

Subtract $x u$ from both sides of the equation.

$1 + u - x u = x$

Step 2.18.4

Subtract $1$ from both sides of the equation.

$u - x u = x - 1$

Step 2.18.5

Factor $u$ out of $u - x u$ .

Tap for more steps...

Step 2.18.5.1

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - x u = x - 1$

Step 2.18.5.2

Factor $u$ out of $- x u$ .

$u \cdot 1 + u (- x) = x - 1$

Step 2.18.5.3

Factor $u$ out of $u \cdot 1 + u (- x)$ .

$u (1 - x) = x - 1$

Step 2.18.6

Divide each term in $u (1 - x) = x - 1$ by $1 - x$ and simplify.

Tap for more steps...

Step 2.18.6.1

Divide each term in $u (1 - x) = x - 1$ by $1 - x$ .

$\frac{u (1 - x)}{1 - x} = \frac{x}{1 - x} + \frac{- 1}{1 - x}$

Step 2.18.6.2

Simplify the left side.

Tap for more steps...

Step 2.18.6.2.1

Cancel the common factor of $1 - x$ .

Tap for more steps...

Step 2.18.6.2.1.1

Cancel the common factor.

$\frac{u (1 - x)}{1 - x} = \frac{x}{1 - x} + \frac{- 1}{1 - x}$

Step 2.18.6.2.1.2

Divide $u$ by $1$ .

$u = \frac{x}{1 - x} + \frac{- 1}{1 - x}$

Step 2.18.6.3

Simplify the right side.

Tap for more steps...

Step 2.18.6.3.1

Combine the numerators over the common denominator.

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

Step 2.18.6.3.2

Cancel the common factor of $x - 1$ and $1 - x$ .

Tap for more steps...

Step 2.18.6.3.2.1

Factor $- 1$ out of $x$ .

$u = \frac{- 1 (- x) - 1}{1 - x}$

Step 2.18.6.3.2.2

Rewrite $- 1$ as $- 1 (1)$ .

$u = \frac{- 1 (- x) - 1 (1)}{1 - x}$

Step 2.18.6.3.2.3

Factor $- 1$ out of $- 1 (- x) - 1 (1)$ .

$u = \frac{- 1 (- x + 1)}{1 - x}$

Step 2.18.6.3.2.4

Reorder terms.

$u = \frac{- 1 (- x + 1)}{- x + 1}$

Step 2.18.6.3.2.5

Cancel the common factor.

$u = \frac{- 1 (- x + 1)}{- x + 1}$

Step 2.18.6.3.2.6

Divide $- 1$ by $1$ .

$u = - 1$

Step 2.18.7

Substitute $\sin (2 y)$ for $u$ .

$\sin (2 y) = - 1$

Step 2.18.8

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

$2 y = \arcsin (- 1)$

Step 2.18.9

Simplify the right side.

Tap for more steps...

Step 2.18.9.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$2 y = - \frac{π}{2}$

Step 2.18.10

Divide each term in $2 y = - \frac{π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 2.18.10.1

Divide each term in $2 y = - \frac{π}{2}$ by $2$ .

$\frac{2 y}{2} = \frac{- \frac{π}{2}}{2}$

Step 2.18.10.2

Simplify the left side.

Tap for more steps...

Step 2.18.10.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.18.10.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- \frac{π}{2}}{2}$

Step 2.18.10.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{π}{2}}{2}$

Step 2.18.10.3

Simplify the right side.

Tap for more steps...

Step 2.18.10.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{π}{2} \cdot \frac{1}{2}$

Step 2.18.10.3.2

Multiply $- \frac{π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 2.18.10.3.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{2}$ .

$y = - \frac{π}{2 \cdot 2}$

Step 2.18.10.3.2.2

Multiply $2$ by $2$ .

$y = - \frac{π}{4}$

Step 2.18.11

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$2 y = 2 π + \frac{π}{2} + π$

Step 2.18.12

Simplify the expression to find the second solution.

Tap for more steps...

Step 2.18.12.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$2 y = 2 π + \frac{π}{2} + π - 2 π$

Step 2.18.12.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$2 y = \frac{3 π}{2}$

Step 2.18.12.3

Divide each term in $2 y = \frac{3 π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 2.18.12.3.1

Divide each term in $2 y = \frac{3 π}{2}$ by $2$ .

$\frac{2 y}{2} = \frac{\frac{3 π}{2}}{2}$

Step 2.18.12.3.2

Simplify the left side.

Tap for more steps...

Step 2.18.12.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.18.12.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\frac{3 π}{2}}{2}$

Step 2.18.12.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{3 π}{2}}{2}$

Step 2.18.12.3.3

Simplify the right side.

Tap for more steps...

Step 2.18.12.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{3 π}{2} \cdot \frac{1}{2}$

Step 2.18.12.3.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 2.18.12.3.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{2}$ .

$y = \frac{3 π}{2 \cdot 2}$

Step 2.18.12.3.3.2.2

Multiply $2$ by $2$ .

$y = \frac{3 π}{4}$

Step 2.18.13

Find the period of $\sin (2 y)$ .

Tap for more steps...

Step 2.18.13.1

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

$\frac{2 π}{| b |}$

Step 2.18.13.2

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

$\frac{2 π}{| 2 |}$

Step 2.18.13.3

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

$\frac{2 π}{2}$

Step 2.18.13.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.18.13.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.18.13.4.2

Divide $π$ by $1$ .

$π$

Step 2.18.14

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

Tap for more steps...

Step 2.18.14.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 2.18.14.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 2.18.14.3

Combine fractions.

Tap for more steps...

Step 2.18.14.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 2.18.14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 2.18.14.4

Simplify the numerator.

Tap for more steps...

Step 2.18.14.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 2.18.14.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 2.18.14.5

List the new angles.

$y = \frac{3 π}{4}$

Step 2.18.15

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

$y = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$

Step 4

Verify if $f^{- 1} (x) = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ is the inverse of $f (x) = \frac{{(\sin (x) + \cos (x))}^{2}}{1 + 2 \sin (x) \cos (x)}$ .

Tap for more steps...

Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{{(\sin (x) + \cos (x))}^{2}}{1 + 2 \sin (x) \cos (x)}$ and $f^{- 1} (x) = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ and compare them.

Step 4.2

Find the range of $f (x) = \frac{{(\sin (x) + \cos (x))}^{2}}{1 + 2 \sin (x) \cos (x)}$ .

Tap for more steps...

Step 4.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, - 1] \cup [1, \infty)$

Step 4.3

Find the domain of $\frac{3 π}{4} + π n$ .

Tap for more steps...

Step 4.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Step 4.4

Since the domain of $f^{- 1} (x) = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ is not equal to the range of $f (x) = \frac{{(\sin (x) + \cos (x))}^{2}}{1 + 2 \sin (x) \cos (x)}$ , then $f^{- 1} (x) = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ is not an inverse of $f (x) = \frac{{(\sin (x) + \cos (x))}^{2}}{1 + 2 \sin (x) \cos (x)}$ .

There is no inverse

Step 5