Trigonometry Examples

$\cos (345)$

Step 1

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

$\cos (15)$

Step 2

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

$\cos (45 - 30)$

Step 3

Separate negation.

$\cos (45 - (30))$

Step 4

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$\cos (45) \cos (30) + \sin (45) \sin (30)$

Step 5

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

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

Step 6

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

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)$

Step 7

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

$\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)$

Step 8

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

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

Step 9

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

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Step 9.1

Simplify each term.

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Step 9.1.1

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

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Step 9.1.1.1

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

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

Step 9.1.1.2

Combine using the product rule for radicals.

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

Step 9.1.1.3

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 9.1.1.4

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 9.1.2

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

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Step 9.1.2.1

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

$\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}$

Step 9.1.2.2

Multiply $2$ by $2$ .

$\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$

Step 9.2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} + \sqrt{2}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6} + \sqrt{2}}{4}$

Decimal Form:

$0.96592582 \dots$