Trigonometry Examples

$f (x) = \arcsin (x) + \arccos (x)$

Step 1

Select a few points to graph.

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Step 1.1

Find the point at $x = - 1$ .

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Step 1.1.1

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

$f (- 1) = \arcsin (- 1) + \arccos (- 1)$

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

$f (- 1) = - \frac{π}{2} + π$

Step 1.1.2.2

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

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

Step 1.1.2.3

Combine fractions.

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Step 1.1.2.3.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 1.1.2.3.2

Combine the numerators over the common denominator.

$f (- 1) = \frac{- π + π \cdot 2}{2}$

Step 1.1.2.4

Simplify the numerator.

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Step 1.1.2.4.1

Move $2$ to the left of $π$ .

$f (- 1) = \frac{- π + 2 \cdot π}{2}$

Step 1.1.2.4.2

Add $- π$ and $2 π$ .

$f (- 1) = \frac{π}{2}$

Step 1.1.2.5

The final answer is $\frac{π}{2}$ .

$\frac{π}{2}$

Step 1.2

List the points in a table.

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

Step 2

The trig function can be graphed using the points.

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

Step 3