Trigonometry Examples

$\cos (\arcsin (- 1) + \frac{π}{4})$

Step 1

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

$\cos (- \frac{π}{2} + \frac{π}{4})$

Step 2

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

$\cos (- \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4})$

Step 3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

$\cos (- \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4})$

Step 3.2

Multiply $2$ by $2$ .

$\cos (- \frac{π \cdot 2}{4} + \frac{π}{4})$

Step 4

Combine the numerators over the common denominator.

$\cos (\frac{- π \cdot 2 + π}{4})$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $- 1$ .

$\cos (\frac{- 2 π + π}{4})$

Step 5.2

Add $- 2 π$ and $π$ .

$\cos (\frac{- π}{4})$

Step 6

Move the negative in front of the fraction.

$\cos (- \frac{π}{4})$

Step 7

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\cos (\frac{7 π}{4})$

Step 8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\cos (\frac{π}{4})$

Step 9

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$