Precalculus Examples

\cos (60 - 45)

$\cos (60 - 45)$

Step 1

Subtract

45

$45$ from

60

$60$ .

\cos (15)

$\cos (15)$

Step 2

The exact value of

\cos (15)

$\cos (15)$ is

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

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

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Step 2.1

Split

15

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

\cos (45 - 30)

$\cos (45 - 30)$

Step 2.2

Separate negation.

\cos (45 - (30))

$\cos (45 - (30))$

Step 2.3

Apply the difference of angles identity

\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)

$\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

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

Step 2.4

The exact value of

\cos (45)

$\cos (45)$ is

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

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

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

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

Step 2.5

The exact value of

\cos (30)

$\cos (30)$ is

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

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

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

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

Step 2.6

The exact value of

\sin (45)

$\sin (45)$ is

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

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

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

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

Step 2.7

The exact value of

\sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 2.8

Simplify

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

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

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Multiply

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

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

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Step 2.8.1.1.1

Multiply

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

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

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

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

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

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

Step 2.8.1.1.2

Combine using the product rule for radicals.

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

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

Step 2.8.1.1.3

Multiply

2

$2$ by

3

$3$ .

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

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

Step 2.8.1.1.4

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 2.8.1.2

Multiply

\frac{\sqrt{2}}{2} \cdot \frac{1}{2}

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

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Step 2.8.1.2.1

Multiply

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

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

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 2.8.1.2.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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

Step 2.8.2

Combine the numerators over the common denominator.

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

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

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

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

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.96592582 \dots

$0.96592582 \dots$