Trigonometry Examples

$\sin (165) = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1

The exact value of $\sin (165)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1

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

$\sin (15) = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.2

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\sin (45 - 30) = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.3

Separate negation.

$\sin (45 - (30)) = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.4

Apply the difference of angles identity.

$\sin (45) \cos (30) - \cos (45) \sin (30) = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.5

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30) = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.6

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

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30) = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.7

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30) = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.8

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.9

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

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

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Step 1.9.1.1.1

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

$\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.9.1.1.2

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.9.1.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.9.1.1.4

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.9.1.2

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

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Step 1.9.1.2.1

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

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2} = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.9.1.2.2

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 1.9.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} = \sin (135) \cos (30) + \cos (135) \sin (30)$

Step 2

Simplify each term.

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Step 2.1

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

$\frac{\sqrt{6} - \sqrt{2}}{4} = \sin (45) \cos (30) + \cos (135) \sin (30)$

Step 2.2

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{2}}{2} \cos (30) + \cos (135) \sin (30)$

Step 2.3

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

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \cos (135) \sin (30)$

Step 2.4

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

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Step 2.4.1

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

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \cos (135) \sin (30)$

Step 2.4.2

Combine using the product rule for radicals.

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \cos (135) \sin (30)$

Step 2.4.3

Multiply $2$ by $3$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{6}}{2 \cdot 2} + \cos (135) \sin (30)$

Step 2.4.4

Multiply $2$ by $2$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{6}}{4} + \cos (135) \sin (30)$

Step 2.5

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.

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{6}}{4} - \cos (45) \sin (30)$

Step 2.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \sin (30)$

Step 2.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 2.8

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

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Step 2.8.1

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

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}$

Step 2.8.2

Multiply $2$ by $2$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$

Step 3