Trigonometry Examples

$\cos (\frac{5 π}{6}) \cos (\frac{2 π}{3}) + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1

Simplify each term.

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Step 1.1

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.

$- \cos (\frac{π}{6}) \cos (\frac{2 π}{3}) + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{3}}{2} \cos (\frac{2 π}{3}) + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1.3

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.

$- \frac{\sqrt{3}}{2} (- \cos (\frac{π}{3})) + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1.4

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

$- \frac{\sqrt{3}}{2} (- \frac{1}{2}) + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1.5

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

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Step 1.5.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{3}}{2} \frac{1}{2} + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1.5.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\frac{\sqrt{3}}{2} \cdot \frac{1}{2} + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1.5.3

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{1}{2}$ .

$\frac{\sqrt{3}}{2 \cdot 2} + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1.5.4

Multiply $2$ by $2$ .

$\frac{\sqrt{3}}{4} + \sin (\frac{5 π}{6}) \sin (\frac{2 π}{3})$

Step 1.6

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

$\frac{\sqrt{3}}{4} + \sin (\frac{π}{6}) \sin (\frac{2 π}{3})$

Step 1.7

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

$\frac{\sqrt{3}}{4} + \frac{1}{2} \sin (\frac{2 π}{3})$

Step 1.8

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

$\frac{\sqrt{3}}{4} + \frac{1}{2} \sin (\frac{π}{3})$

Step 1.9

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

$\frac{\sqrt{3}}{4} + \frac{1}{2} \cdot \frac{\sqrt{3}}{2}$

Step 1.10

Multiply $\frac{1}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.10.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{2 \cdot 2}$

Step 1.10.2

Multiply $2$ by $2$ .

$\frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{\sqrt{3} + \sqrt{3}}{4}$

Step 2.2

Add $\sqrt{3}$ and $\sqrt{3}$ .

$\frac{2 \sqrt{3}}{4}$

Step 2.3

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

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Step 2.3.1

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

$\frac{2 (\sqrt{3})}{4}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $4$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$