Trigonometry Examples

$f (t) = - \frac{1}{4} \cdot \cos (2 t)$

Step 1

Write $f (t) = - \frac{1}{4} \cdot \cos (2 t)$ as an equation.

$y = - \frac{1}{4} \cdot \cos (2 t)$

Step 2

Interchange the variables.

$t = - \frac{1}{4} \cdot \cos (2 y)$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $- \frac{1}{4} \cdot \cos (2 y) = t$ .

$- \frac{1}{4} \cdot \cos (2 y) = t$

Step 3.2

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{1}{4} \cos (2 y)) = - 4 t$

Step 3.3

Simplify the left side.

Tap for more steps...

Step 3.3.1

Simplify $- 4 (- \frac{1}{4} \cos (2 y))$ .

Tap for more steps...

Step 3.3.1.1

Combine $\cos (2 y)$ and $\frac{1}{4}$ .

$- 4 (- \frac{\cos (2 y)}{4}) = - 4 t$

Step 3.3.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.1.2.1

Move the leading negative in $- \frac{\cos (2 y)}{4}$ into the numerator.

$- 4 \frac{- \cos (2 y)}{4} = - 4 t$

Step 3.3.1.2.2

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{- \cos (2 y)}{4} = - 4 t$

Step 3.3.1.2.3

Cancel the common factor.

$4 \cdot - 1 \frac{- \cos (2 y)}{4} = - 4 t$

Step 3.3.1.2.4

Rewrite the expression.

$- - \cos (2 y) = - 4 t$

Step 3.3.1.3

Multiply.

Tap for more steps...

Step 3.3.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \cos (2 y) = - 4 t$

Step 3.3.1.3.2

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

$\cos (2 y) = - 4 t$

Step 3.4

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

$2 y = \arccos (- 4 t)$

Step 3.5

Divide each term in $2 y = \arccos (- 4 t)$ by $2$ and simplify.

Tap for more steps...

Step 3.5.1

Divide each term in $2 y = \arccos (- 4 t)$ by $2$ .

$\frac{2 y}{2} = \frac{\arccos (- 4 t)}{2}$

Step 3.5.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\arccos (- 4 t)}{2}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (- 4 t)}{2}$

Step 4

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

$f^{- 1} (t) = \frac{\arccos (- 4 t)}{2}$

Step 5

Verify if $f^{- 1} (t) = \frac{\arccos (- 4 t)}{2}$ is the inverse of $f (t) = - \frac{1}{4} \cdot \cos (2 t)$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (- \frac{1}{4} \cdot \cos (2 t))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{1}{4} \cdot \cos (2 t)) = \frac{\arccos (- 4 (- \frac{1}{4} \cdot \cos (2 t)))}{2}$

Step 5.2.3

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1

Combine $\cos (2 t)$ and $\frac{1}{4}$ .

$f^{- 1} (- \frac{1}{4} \cdot \cos (2 t)) = \frac{\arccos (- 4 (- \frac{\cos (2 t)}{4}))}{2}$

Step 5.2.3.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.2.3.2.1

Move the leading negative in $- \frac{\cos (2 t)}{4}$ into the numerator.

$f^{- 1} (- \frac{1}{4} \cdot \cos (2 t)) = \frac{\arccos (- 4 \frac{- \cos (2 t)}{4})}{2}$

Step 5.2.3.2.2

Factor $4$ out of $- 4$ .

$f^{- 1} (- \frac{1}{4} \cdot \cos (2 t)) = \frac{\arccos (4 (- 1) (\frac{- \cos (2 t)}{4}))}{2}$

Step 5.2.3.2.3

Cancel the common factor.

$f^{- 1} (- \frac{1}{4} \cdot \cos (2 t)) = \frac{\arccos (4 \cdot (- 1 \frac{- \cos (2 t)}{4}))}{2}$

Step 5.2.3.2.4

Rewrite the expression.

$f^{- 1} (- \frac{1}{4} \cdot \cos (2 t)) = \frac{\arccos (- 1 (- \cos (2 t)))}{2}$

Step 5.2.3.3

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{1}{4} \cdot \cos (2 t)) = \frac{\arccos (1 \cos (2 t))}{2}$

Step 5.2.3.4

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

$f^{- 1} (- \frac{1}{4} \cdot \cos (2 t)) = \frac{\arccos (\cos (2 t))}{2}$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{\arccos (- 4 t)}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (- 4 t)}{2}) = - \frac{1}{4} \cdot \cos (2 (\frac{\arccos (- 4 t)}{2}))$

Step 5.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.3.1

Cancel the common factor.

$f (\frac{\arccos (- 4 t)}{2}) = - \frac{1}{4} \cdot \cos (2 (\frac{\arccos (- 4 t)}{2}))$

Step 5.3.3.2

Rewrite the expression.

$f (\frac{\arccos (- 4 t)}{2}) = - \frac{1}{4} \cdot \cos (\arccos (- 4 t))$

Step 5.3.4

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (- 4 t)}{2}) = - \frac{1}{4} \cdot (- 4 t)$

Step 5.3.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.3.5.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$f (\frac{\arccos (- 4 t)}{2}) = \frac{- 1}{4} \cdot (- 4 t)$

Step 5.3.5.2

Factor $4$ out of $- 4 t$ .

$f (\frac{\arccos (- 4 t)}{2}) = \frac{- 1}{4} \cdot (4 (- t))$

Step 5.3.5.3

Cancel the common factor.

$f (\frac{\arccos (- 4 t)}{2}) = \frac{- 1}{4} \cdot (4 (- t))$

Step 5.3.5.4

Rewrite the expression.

$f (\frac{\arccos (- 4 t)}{2}) = - 1 \cdot (- t)$

Step 5.3.6

Multiply.

Tap for more steps...

Step 5.3.6.1

Multiply $- 1$ by $- 1$ .

$f (\frac{\arccos (- 4 t)}{2}) = 1 \cdot t$

Step 5.3.6.2

Multiply $t$ by $1$ .

$f (\frac{\arccos (- 4 t)}{2}) = t$

Step 5.4

Since $f^{- 1} (f (t)) = t$ and $f (f^{- 1} (t)) = t$ , then $f^{- 1} (t) = \frac{\arccos (- 4 t)}{2}$ is the inverse of $f (t) = - \frac{1}{4} \cdot \cos (2 t)$ .

$f^{- 1} (t) = \frac{\arccos (- 4 t)}{2}$