Trigonometry Examples

$\arccos (\frac{1}{4} x) - \frac{π}{2}$

Step 1

Select a few points to graph.

Tap for more steps...

Step 1.1

Find the point at $x = - 4$ .

Tap for more steps...

Step 1.1.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 1.1.2

Simplify the result.

Tap for more steps...

Step 1.1.2.1

Simplify each term.

Tap for more steps...

Step 1.1.2.1.1

Divide $- 4$ by $4$ .

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

Step 1.1.2.1.2

The exact value of $\arccos (- 1)$ is $π$ .

$f (- 4) = π - \frac{π}{2}$

Step 1.1.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 4) = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.1.2.3

Combine fractions.

Tap for more steps...

Step 1.1.2.3.1

Combine $π$ and $\frac{2}{2}$ .

$f (- 4) = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 1.1.2.3.2

Combine the numerators over the common denominator.

$f (- 4) = \frac{π \cdot 2 - π}{2}$

Step 1.1.2.4

Simplify the numerator.

Tap for more steps...

Step 1.1.2.4.1

Move $2$ to the left of $π$ .

$f (- 4) = \frac{2 \cdot π - π}{2}$

Step 1.1.2.4.2

Subtract $π$ from $2 π$ .

$f (- 4) = \frac{π}{2}$

Step 1.1.2.5

The final answer is $\frac{π}{2}$ .

$\frac{π}{2}$

Step 1.2

Find the point at $x = - 2$ .

Tap for more steps...

Step 1.2.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 1.2.2

Simplify the result.

Tap for more steps...

Step 1.2.2.1

Simplify each term.

Tap for more steps...

Step 1.2.2.1.1

Cancel the common factor of $- 2$ and $4$ .

Tap for more steps...

Step 1.2.2.1.1.1

Factor $2$ out of $- 2$ .

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

Step 1.2.2.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.2.1.1.2.1

Factor $2$ out of $4$ .

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

Step 1.2.2.1.1.2.2

Cancel the common factor.

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

Step 1.2.2.1.1.2.3

Rewrite the expression.

$f (- 2) = \arccos (\frac{- 1}{2}) - \frac{π}{2}$

Step 1.2.2.1.2

Move the negative in front of the fraction.

$f (- 2) = \arccos (- \frac{1}{2}) - \frac{π}{2}$

Step 1.2.2.1.3

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$f (- 2) = \frac{2 π}{3} - \frac{π}{2}$

Step 1.2.2.2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 2) = \frac{2 π}{3} \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.2.2.3

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 2) = \frac{2 π}{3} \cdot \frac{2}{2} - \frac{π}{2} \cdot \frac{3}{3}$

Step 1.2.2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.2.2.4.1

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

$f (- 2) = \frac{2 π \cdot 2}{3 \cdot 2} - \frac{π}{2} \cdot \frac{3}{3}$

Step 1.2.2.4.2

Multiply $3$ by $2$ .

$f (- 2) = \frac{2 π \cdot 2}{6} - \frac{π}{2} \cdot \frac{3}{3}$

Step 1.2.2.4.3

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

$f (- 2) = \frac{2 π \cdot 2}{6} - \frac{π \cdot 3}{2 \cdot 3}$

Step 1.2.2.4.4

Multiply $2$ by $3$ .

$f (- 2) = \frac{2 π \cdot 2}{6} - \frac{π \cdot 3}{6}$

Step 1.2.2.5

Combine the numerators over the common denominator.

$f (- 2) = \frac{2 π \cdot 2 - π \cdot 3}{6}$

Step 1.2.2.6

Simplify the numerator.

Tap for more steps...

Step 1.2.2.6.1

Multiply $2$ by $2$ .

$f (- 2) = \frac{4 π - π \cdot 3}{6}$

Step 1.2.2.6.2

Multiply $3$ by $- 1$ .

$f (- 2) = \frac{4 π - 3 π}{6}$

Step 1.2.2.6.3

Subtract $3 π$ from $4 π$ .

$f (- 2) = \frac{π}{6}$

Step 1.2.2.7

The final answer is $\frac{π}{6}$ .

$\frac{π}{6}$

Step 1.3

Find the point at $x = 0$ .

Tap for more steps...

Step 1.3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \arccos (\frac{0}{4}) - \frac{π}{2}$

Step 1.3.2

Simplify the result.

Tap for more steps...

Step 1.3.2.1

Simplify each term.

Tap for more steps...

Step 1.3.2.1.1

Divide $0$ by $4$ .

$f (0) = \arccos (0) - \frac{π}{2}$

Step 1.3.2.1.2

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$f (0) = \frac{π}{2} - \frac{π}{2}$

Step 1.3.2.2

Simplify terms.

Tap for more steps...

Step 1.3.2.2.1

Combine the numerators over the common denominator.

$f (0) = \frac{π - π}{2}$

Step 1.3.2.2.2

Subtract $π$ from $π$ .

$f (0) = \frac{0}{2}$

Step 1.3.2.2.3

Divide $0$ by $2$ .

$f (0) = 0$

Step 1.3.2.3

The final answer is $0$ .

$0$

Step 1.4

Find the point at $x = 2$ .

Tap for more steps...

Step 1.4.1

Replace the variable $x$ with $2$ in the expression.

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

Step 1.4.2

Simplify the result.

Tap for more steps...

Step 1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.1.1

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 1.4.2.1.1.1

Factor $2$ out of $2$ .

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

Step 1.4.2.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.4.2.1.1.2.1

Factor $2$ out of $4$ .

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

Step 1.4.2.1.1.2.2

Cancel the common factor.

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

Step 1.4.2.1.1.2.3

Rewrite the expression.

$f (2) = \arccos (\frac{1}{2}) - \frac{π}{2}$

Step 1.4.2.1.2

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$f (2) = \frac{π}{3} - \frac{π}{2}$

Step 1.4.2.2

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (2) = \frac{π}{3} \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.4.2.3

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (2) = \frac{π}{3} \cdot \frac{2}{2} - \frac{π}{2} \cdot \frac{3}{3}$

Step 1.4.2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.4.2.4.1

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

$f (2) = \frac{π \cdot 2}{3 \cdot 2} - \frac{π}{2} \cdot \frac{3}{3}$

Step 1.4.2.4.2

Multiply $3$ by $2$ .

$f (2) = \frac{π \cdot 2}{6} - \frac{π}{2} \cdot \frac{3}{3}$

Step 1.4.2.4.3

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

$f (2) = \frac{π \cdot 2}{6} - \frac{π \cdot 3}{2 \cdot 3}$

Step 1.4.2.4.4

Multiply $2$ by $3$ .

$f (2) = \frac{π \cdot 2}{6} - \frac{π \cdot 3}{6}$

Step 1.4.2.5

Combine the numerators over the common denominator.

$f (2) = \frac{π \cdot 2 - π \cdot 3}{6}$

Step 1.4.2.6

Simplify the numerator.

Tap for more steps...

Step 1.4.2.6.1

Move $2$ to the left of $π$ .

$f (2) = \frac{2 \cdot π - π \cdot 3}{6}$

Step 1.4.2.6.2

Multiply $3$ by $- 1$ .

$f (2) = \frac{2 π - 3 π}{6}$

Step 1.4.2.6.3

Subtract $3 π$ from $2 π$ .

$f (2) = \frac{- π}{6}$

Step 1.4.2.7

Move the negative in front of the fraction.

$f (2) = - \frac{π}{6}$

Step 1.4.2.8

The final answer is $- \frac{π}{6}$ .

$- \frac{π}{6}$

Step 1.5

Find the point at $x = 4$ .

Tap for more steps...

Step 1.5.1

Replace the variable $x$ with $4$ in the expression.

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

Step 1.5.2

Simplify the result.

Tap for more steps...

Step 1.5.2.1

Simplify each term.

Tap for more steps...

Step 1.5.2.1.1

Divide $4$ by $4$ .

$f (4) = \arccos (1) - \frac{π}{2}$

Step 1.5.2.1.2

The exact value of $\arccos (1)$ is $0$ .

$f (4) = 0 - \frac{π}{2}$

Step 1.5.2.2

Subtract $\frac{π}{2}$ from $0$ .

$f (4) = - \frac{π}{2}$

Step 1.5.2.3

The final answer is $- \frac{π}{2}$ .

$- \frac{π}{2}$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 4 & \frac{π}{2} \\ - 2 & \frac{π}{6} \\ 0 & 0 \\ 2 & - \frac{π}{6} \\ 4 & - \frac{π}{2} \end{array}$

Step 2

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ - 4 & \frac{π}{2} \\ - 2 & \frac{π}{6} \\ 0 & 0 \\ 2 & - \frac{π}{6} \\ 4 & - \frac{π}{2} \end{array}$

Step 3