Trigonometry Examples

${\cos (\frac{150}{400})}^{a}$

Step 1

Interchange the variables.

$a = {\cos (\frac{150}{400})}^{y}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as ${\cos (\frac{150}{400})}^{y} = a$ .

${\cos (\frac{150}{400})}^{y} = a$

Step 2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({\cos (\frac{150}{400})}^{y}) = \ln (a)$

Step 2.3

Expand the left side.

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Step 2.3.1

Expand $\ln ({\cos (\frac{150}{400})}^{y})$ by moving $y$ outside the logarithm.

$y \ln (\cos (\frac{150}{400})) = \ln (a)$

Step 2.3.2

Reduce the expression $\frac{150}{400}$ by cancelling the common factors.

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Step 2.3.2.1

Factor $50$ out of $150$ .

$y \ln (\cos (\frac{50 (3)}{400})) = \ln (a)$

Step 2.3.2.2

Factor $50$ out of $400$ .

$y \ln (\cos (\frac{50 \cdot 3}{50 \cdot 8})) = \ln (a)$

Step 2.3.2.3

Cancel the common factor.

$y \ln (\cos (\frac{50 \cdot 3}{50 \cdot 8})) = \ln (a)$

Step 2.3.2.4

Rewrite the expression.

$y \ln (\cos (\frac{3}{8})) = \ln (a)$

Step 2.4

Simplify the left side.

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Step 2.4.1

Evaluate $\cos (\frac{3}{8})$ .

$y \ln (0.93050762) = \ln (a)$

Step 2.5

Divide each term in $y \ln (0.93050762) = \ln (a)$ by $\ln (0.93050762)$ and simplify.

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Step 2.5.1

Divide each term in $y \ln (0.93050762) = \ln (a)$ by $\ln (0.93050762)$ .

$\frac{y \ln (0.93050762)}{\ln (0.93050762)} = \frac{\ln (a)}{\ln (0.93050762)}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $\ln (0.93050762)$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{y \ln (0.93050762)}{\ln (0.93050762)} = \frac{\ln (a)}{\ln (0.93050762)}$

Step 2.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (a)}{\ln (0.93050762)}$

Step 3

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

$f^{- 1} (a) = \frac{\ln (a)}{\ln (0.93050762)}$

Step 4

Verify if $f^{- 1} (a) = \frac{\ln (a)}{\ln (0.93050762)}$ is the inverse of $f (a) = {\cos (\frac{150}{400})}^{a}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} ({\cos (\frac{150}{400})}^{a})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = \frac{\ln ({\cos (\frac{150}{400})}^{a})}{\ln (0.93050762)}$

Step 4.2.3

Expand $\ln ({\cos (\frac{150}{400})}^{a})$ by moving $a$ outside the logarithm.

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = \frac{a \ln (\cos (\frac{150}{400}))}{\ln (0.93050762)}$

Step 4.2.4

Cancel the common factor of $150$ and $400$ .

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Step 4.2.4.1

Factor $50$ out of $150$ .

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = \frac{a \ln (\cos (\frac{50 (3)}{400}))}{\ln (0.93050762)}$

Step 4.2.4.2

Cancel the common factors.

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Step 4.2.4.2.1

Factor $50$ out of $400$ .

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = \frac{a \ln (\cos (\frac{50 \cdot 3}{50 \cdot 8}))}{\ln (0.93050762)}$

Step 4.2.4.2.2

Cancel the common factor.

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = \frac{a \ln (\cos (\frac{50 \cdot 3}{50 \cdot 8}))}{\ln (0.93050762)}$

Step 4.2.4.2.3

Rewrite the expression.

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = \frac{a \ln (\cos (\frac{3}{8}))}{\ln (0.93050762)}$

Step 4.2.5

Evaluate $\cos (\frac{3}{8})$ .

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = \frac{a \ln (0.93050762)}{\ln (0.93050762)}$

Step 4.2.6

Cancel the common factor of $\ln (0.93050762)$ .

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Step 4.2.6.1

Cancel the common factor.

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = \frac{a \ln (0.93050762)}{\ln (0.93050762)}$

Step 4.2.6.2

Divide $a$ by $1$ .

$f^{- 1} ({\cos (\frac{150}{400})}^{a}) = a$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{\ln (a)}{\ln (0.93050762)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (a)}{\ln (0.93050762)}) = {\cos (\frac{150}{400})}^{\frac{\ln (a)}{\ln (0.93050762)}}$

Step 4.3.3

Cancel the common factor of $150$ and $400$ .

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Step 4.3.3.1

Factor $50$ out of $150$ .

$f (\frac{\ln (a)}{\ln (0.93050762)}) = {\cos (\frac{50 (3)}{400})}^{\frac{\ln (a)}{\ln (0.93050762)}}$

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $50$ out of $400$ .

$f (\frac{\ln (a)}{\ln (0.93050762)}) = {\cos (\frac{50 \cdot 3}{50 \cdot 8})}^{\frac{\ln (a)}{\ln (0.93050762)}}$

Step 4.3.3.2.2

Cancel the common factor.

$f (\frac{\ln (a)}{\ln (0.93050762)}) = {\cos (\frac{50 \cdot 3}{50 \cdot 8})}^{\frac{\ln (a)}{\ln (0.93050762)}}$

Step 4.3.3.2.3

Rewrite the expression.

$f (\frac{\ln (a)}{\ln (0.93050762)}) = {\cos (\frac{3}{8})}^{\frac{\ln (a)}{\ln (0.93050762)}}$

Step 4.3.4

Evaluate $\cos (\frac{3}{8})$ .

$f (\frac{\ln (a)}{\ln (0.93050762)}) = {0.93050762}^{\frac{\ln (a)}{\ln (0.93050762)}}$

Step 4.3.5

Use the change of base rule $\frac{\log_{b} (x)}{\log_{b} (a)} = \log_{a} (x)$ .

$f (\frac{\ln (a)}{\ln (0.93050762)}) = {0.93050762}^{\log_{0.93050762} (a)}$

Step 4.3.6

Exponentiation and log are inverse functions.

$f (\frac{\ln (a)}{\ln (0.93050762)}) = a$

Step 4.4

Since $f^{- 1} (f (a)) = a$ and $f (f^{- 1} (a)) = a$ , then $f^{- 1} (a) = \frac{\ln (a)}{\ln (0.93050762)}$ is the inverse of $f (a) = {\cos (\frac{150}{400})}^{a}$ .

$f^{- 1} (a) = \frac{\ln (a)}{\ln (0.93050762)}$