Trigonometry Examples

$\cos (4 x) - \sin (4 x)$

Step 1

Interchange the variables.

$x = \cos (4 y) - \sin (4 y)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\cos (4 y) - \sin (4 y) = x$ .

$\cos (4 y) - \sin (4 y) = x$

Step 2.2

Use the identity to solve the equation. In this identity, $θ$ represents the angle created by plotting point $(a, b)$ on a graph and therefore can be found using $θ = \arctan (\frac{b}{a})$ .

$a \sin (x) + b \cos (x) = R \sin (x + θ)$ where $R = \sqrt{a^{2} + b^{2}}$ and $θ = \arctan (\frac{b}{a})$

Step 2.3

Set up the equation to find the value of $θ$ .

$\arctan (\frac{1}{- 1})$

Step 2.4

Take the inverse tangent to solve the equation for $θ$ .

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Step 2.4.1

Divide $1$ by $- 1$ .

$θ = \arctan (- 1)$

Step 2.4.2

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$θ = - \frac{π}{4}$

Step 2.5

Solve to find the value of $R$ .

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Step 2.5.1

Raise $- 1$ to the power of $2$ .

$R = \sqrt{1 + {(1)}^{2}}$

Step 2.5.2

One to any power is one.

$R = \sqrt{1 + 1}$

Step 2.5.3

Add $1$ and $1$ .

$R = \sqrt{2}$

Step 2.6

Substitute the known values into the equation.

$(\sqrt{2}) \sin (4 y - \frac{π}{4}) = x$

Step 2.7

Divide each term in $\sqrt{2} \sin (4 y - \frac{π}{4}) = x$ by $\sqrt{2}$ and simplify.

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Step 2.7.1

Divide each term in $\sqrt{2} \sin (4 y - \frac{π}{4}) = x$ by $\sqrt{2}$ .

$\frac{\sqrt{2} \sin (4 y - \frac{π}{4})}{\sqrt{2}} = \frac{x}{\sqrt{2}}$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $\sqrt{2}$ .

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{\sqrt{2} \sin (4 y - \frac{π}{4})}{\sqrt{2}} = \frac{x}{\sqrt{2}}$

Step 2.7.2.1.2

Divide $\sin (4 y - \frac{π}{4})$ by $1$ .

$\sin (4 y - \frac{π}{4}) = \frac{x}{\sqrt{2}}$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

Multiply $\frac{x}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (4 y - \frac{π}{4}) = \frac{x}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.7.3.2

Combine and simplify the denominator.

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Step 2.7.3.2.1

Multiply $\frac{x}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.7.3.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.7.3.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.7.3.2.4

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

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.7.3.2.5

Add $1$ and $1$ .

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.7.3.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.7.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.7.3.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.7.3.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.7.3.2.6.4

Cancel the common factor of $2$ .

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Step 2.7.3.2.6.4.1

Cancel the common factor.

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.7.3.2.6.4.2

Rewrite the expression.

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{2^{1}}$

Step 2.7.3.2.6.5

Evaluate the exponent.

$\sin (4 y - \frac{π}{4}) = \frac{x \sqrt{2}}{2}$

Step 2.8

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

$4 y - \frac{π}{4} = \arcsin (\frac{x \sqrt{2}}{2})$

Step 2.9

Add $\frac{π}{4}$ to both sides of the equation.

$4 y = \arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}$

Step 2.10

Divide each term in $4 y = \arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}$ by $4$ and simplify.

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Step 2.10.1

Divide each term in $4 y = \arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}$ by $4$ .

$\frac{4 y}{4} = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{\frac{π}{4}}{4}$

Step 2.10.2

Simplify the left side.

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Step 2.10.2.1

Cancel the common factor of $4$ .

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Step 2.10.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{\frac{π}{4}}{4}$

Step 2.10.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{\frac{π}{4}}{4}$

Step 2.10.3

Simplify the right side.

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Step 2.10.3.1

Simplify each term.

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Step 2.10.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{4} \cdot \frac{1}{4}$

Step 2.10.3.1.2

Multiply $\frac{π}{4} \cdot \frac{1}{4}$ .

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Step 2.10.3.1.2.1

Multiply $\frac{π}{4}$ by $\frac{1}{4}$ .

$y = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{4 \cdot 4}$

Step 2.10.3.1.2.2

Multiply $4$ by $4$ .

$y = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}$

Step 3

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

$f^{- 1} (x) = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}$ is the inverse of $f (x) = \cos (4 x) - \sin (4 x)$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\cos (4 x) - \sin (4 x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\cos (4 x) - \sin (4 x)) = \frac{\arcsin (\frac{(\cos (4 x) - \sin (4 x)) \sqrt{2}}{2})}{4} + \frac{π}{16}$

Step 4.2.3

Reorder factors in $\frac{\arcsin (\frac{(\cos (4 x) - \sin (4 x)) \sqrt{2}}{2})}{4} + \frac{π}{16}$ .

$f^{- 1} (\cos (4 x) - \sin (4 x)) = \frac{\arcsin (\frac{\sqrt{2} (\cos (4 x) - \sin (4 x))}{2})}{4} + \frac{π}{16}$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16})) - \sin (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}))$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4}) + 4 (\frac{π}{16})) - \sin (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}))$

Step 4.3.3.2

Cancel the common factor of $4$ .

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Step 4.3.3.2.1

Cancel the common factor.

Step 4.3.3.2.2

Rewrite the expression.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + 4 (\frac{π}{16})) - \sin (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}))$

Step 4.3.3.3

Cancel the common factor of $4$ .

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Step 4.3.3.3.1

Factor $4$ out of $16$ .

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + 4 (\frac{π}{4 (4)})) - \sin (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}))$

Step 4.3.3.3.2

Cancel the common factor.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + 4 (\frac{π}{4 \cdot 4})) - \sin (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}))$

Step 4.3.3.3.3

Rewrite the expression.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}) - \sin (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}))$

Step 4.3.3.4

Apply the distributive property.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}) - \sin (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4}) + 4 (\frac{π}{16}))$

Step 4.3.3.5

Cancel the common factor of $4$ .

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Step 4.3.3.5.1

Cancel the common factor.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}) - \sin (4 (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4}) + 4 (\frac{π}{16}))$

Step 4.3.3.5.2

Rewrite the expression.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}) - \sin (\arcsin (\frac{x \sqrt{2}}{2}) + 4 (\frac{π}{16}))$

Step 4.3.3.6

Cancel the common factor of $4$ .

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Step 4.3.3.6.1

Factor $4$ out of $16$ .

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}) - \sin (\arcsin (\frac{x \sqrt{2}}{2}) + 4 (\frac{π}{4 (4)}))$

Step 4.3.3.6.2

Cancel the common factor.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}) - \sin (\arcsin (\frac{x \sqrt{2}}{2}) + 4 (\frac{π}{4 \cdot 4}))$

Step 4.3.3.6.3

Rewrite the expression.

$f (\frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}) = \cos (\arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4}) - \sin (\arcsin (\frac{x \sqrt{2}}{2}) + \frac{π}{4})$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}$ is the inverse of $f (x) = \cos (4 x) - \sin (4 x)$ .

$f^{- 1} (x) = \frac{\arcsin (\frac{x \sqrt{2}}{2})}{4} + \frac{π}{16}$