Trigonometry Examples

$y = \arcsin (x) + \frac{π}{2}$

Step 1

Select a few points to graph.

Tap for more steps...

Step 1.1

Find the point at $x = - 1$ .

Tap for more steps...

Step 1.1.1

Replace the variable $x$ with $- 1$ in the expression.

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

Step 1.1.2

Simplify the result.

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

Combine the numerators over the common denominator.

$f (- 1) = \frac{- π + π}{2}$

Step 1.1.2.3

Add $- π$ and $π$ .

$f (- 1) = \frac{0}{2}$

Step 1.1.2.4

Divide $0$ by $2$ .

$f (- 1) = 0$

Step 1.1.2.5

The final answer is $0$ .

$0$

Step 1.2

Find the point at $x = - \frac{1}{2}$ .

Tap for more steps...

Step 1.2.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

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

Step 1.2.2

Simplify the result.

Tap for more steps...

Step 1.2.2.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- \frac{π}{6}$ .

$f (- \frac{1}{2}) = - \frac{π}{6} + \frac{π}{2}$

Step 1.2.2.2

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

$f (- \frac{1}{2}) = - \frac{π}{6} + \frac{π}{2} \cdot \frac{3}{3}$

Step 1.2.2.3

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

Tap for more steps...

Step 1.2.2.3.1

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

$f (- \frac{1}{2}) = - \frac{π}{6} + \frac{π \cdot 3}{2 \cdot 3}$

Step 1.2.2.3.2

Multiply $2$ by $3$ .

$f (- \frac{1}{2}) = - \frac{π}{6} + \frac{π \cdot 3}{6}$

Step 1.2.2.4

Combine the numerators over the common denominator.

$f (- \frac{1}{2}) = \frac{- π + π \cdot 3}{6}$

Step 1.2.2.5

Simplify the numerator.

Tap for more steps...

Step 1.2.2.5.1

Move $3$ to the left of $π$ .

$f (- \frac{1}{2}) = \frac{- π + 3 \cdot π}{6}$

Step 1.2.2.5.2

Add $- π$ and $3 π$ .

$f (- \frac{1}{2}) = \frac{2 π}{6}$

Step 1.2.2.6

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 1.2.2.6.1

Factor $2$ out of $2 π$ .

$f (- \frac{1}{2}) = \frac{2 (π)}{6}$

Step 1.2.2.6.2

Cancel the common factors.

Tap for more steps...

Step 1.2.2.6.2.1

Factor $2$ out of $6$ .

$f (- \frac{1}{2}) = \frac{2 π}{2 \cdot 3}$

Step 1.2.2.6.2.2

Cancel the common factor.

$f (- \frac{1}{2}) = \frac{2 π}{2 \cdot 3}$

Step 1.2.2.6.2.3

Rewrite the expression.

$f (- \frac{1}{2}) = \frac{π}{3}$

Step 1.2.2.7

The final answer is $\frac{π}{3}$ .

$\frac{π}{3}$

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.

$f (0) = \arcsin (0) + \frac{π}{2}$

Step 1.3.2

Simplify the result.

Tap for more steps...

Step 1.3.2.1

The exact value of $\arcsin (0)$ is $0$ .

$f (0) = 0 + \frac{π}{2}$

Step 1.3.2.2

Add $0$ and $\frac{π}{2}$ .

$f (0) = \frac{π}{2}$

Step 1.3.2.3

The final answer is $\frac{π}{2}$ .

$\frac{π}{2}$

Step 1.4

Find the point at $x = \frac{1}{2}$ .

Tap for more steps...

Step 1.4.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = \arcsin (\frac{1}{2}) + \frac{π}{2}$

Step 1.4.2

Simplify the result.

Tap for more steps...

Step 1.4.2.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$f (\frac{1}{2}) = \frac{π}{6} + \frac{π}{2}$

Step 1.4.2.2

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

$f (\frac{1}{2}) = \frac{π}{6} + \frac{π}{2} \cdot \frac{3}{3}$

Step 1.4.2.3

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

Tap for more steps...

Step 1.4.2.3.1

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

$f (\frac{1}{2}) = \frac{π}{6} + \frac{π \cdot 3}{2 \cdot 3}$

Step 1.4.2.3.2

Multiply $2$ by $3$ .

$f (\frac{1}{2}) = \frac{π}{6} + \frac{π \cdot 3}{6}$

Step 1.4.2.4

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = \frac{π + π \cdot 3}{6}$

Step 1.4.2.5

Simplify the numerator.

Tap for more steps...

Step 1.4.2.5.1

Move $3$ to the left of $π$ .

$f (\frac{1}{2}) = \frac{π + 3 \cdot π}{6}$

Step 1.4.2.5.2

Add $π$ and $3 π$ .

$f (\frac{1}{2}) = \frac{4 π}{6}$

Step 1.4.2.6

Cancel the common factor of $4$ and $6$ .

Tap for more steps...

Step 1.4.2.6.1

Factor $2$ out of $4 π$ .

$f (\frac{1}{2}) = \frac{2 (2 π)}{6}$

Step 1.4.2.6.2

Cancel the common factors.

Tap for more steps...

Step 1.4.2.6.2.1

Factor $2$ out of $6$ .

$f (\frac{1}{2}) = \frac{2 (2 π)}{2 (3)}$

Step 1.4.2.6.2.2

Cancel the common factor.

$f (\frac{1}{2}) = \frac{2 (2 π)}{2 \cdot 3}$

Step 1.4.2.6.2.3

Rewrite the expression.

$f (\frac{1}{2}) = \frac{2 π}{3}$

Step 1.4.2.7

The final answer is $\frac{2 π}{3}$ .

$\frac{2 π}{3}$

Step 1.5

Find the point at $x = 1$ .

Tap for more steps...

Step 1.5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \arcsin (1) + \frac{π}{2}$

Step 1.5.2

Simplify the result.

Tap for more steps...

Step 1.5.2.1

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

$f (1) = \frac{π}{2} + \frac{π}{2}$

Step 1.5.2.2

Combine the numerators over the common denominator.

$f (1) = \frac{π + π}{2}$

Step 1.5.2.3

Add $π$ and $π$ .

$f (1) = \frac{2 π}{2}$

Step 1.5.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.2.4.1

Cancel the common factor.

$f (1) = \frac{2 π}{2}$

Step 1.5.2.4.2

Divide $π$ by $1$ .

$f (1) = π$

Step 1.5.2.5

The final answer is $π$ .

$π$

Step 1.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - 1 & 0 \\ - \frac{1}{2} & \frac{π}{3} \\ 0 & \frac{π}{2} \\ \frac{1}{2} & \frac{2 π}{3} \\ 1 & π \end{array}$

Step 2

The trig function can be graphed using the points.

$\begin{array}{c} x & f (x) \\ - 1 & 0 \\ - \frac{1}{2} & \frac{π}{3} \\ 0 & \frac{π}{2} \\ \frac{1}{2} & \frac{2 π}{3} \\ 1 & π \end{array}$

Step 3