Trigonometry Examples

$\cos (870) - \sin (780)$

Step 1

Simplify each term.

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Step 1.1

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\cos (150) - \sin (780)$

Step 1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \cos (30) - \sin (780)$

Step 1.3

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{3}}{2} - \sin (780)$

Step 1.4

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$- \frac{\sqrt{3}}{2} - \sin (60)$

Step 1.5

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{- \sqrt{3} - \sqrt{3}}{2}$

Step 2.2

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$\frac{- 2 \sqrt{3}}{2}$

Step 2.3

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

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Step 2.3.1

Factor $2$ out of $- 2 \sqrt{3}$ .

$\frac{2 (- \sqrt{3})}{2}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (- \sqrt{3})}{2 (1)}$

Step 2.3.2.2

Cancel the common factor.

$\frac{2 (- \sqrt{3})}{2 \cdot 1}$

Step 2.3.2.3

Rewrite the expression.

$\frac{- \sqrt{3}}{1}$

Step 2.3.2.4

Divide $- \sqrt{3}$ by $1$ .

$- \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{3}$

Decimal Form:

$- 1.73205080 \dots$