Trigonometry Examples

cos (45 - 30)

$\cos (45 - 30)$

Step 1

Use the difference formula for cosine to simplify the expression. The formula states that

cos (A - B) = cos (A) cos (B) + sin (A) sin (B)

$\cos (A - B) = \cos (A) \cos (B) + \sin (A) \sin (B)$ .

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

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

Step 2

Remove parentheses.

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

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

Step 3

Simplify each term.

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Step 3.1

The exact value of

cos (45)

$\cos (45)$ is

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

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

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

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

Step 3.2

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) \cdot sin (30)

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

Step 3.3

Multiply

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

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

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Step 3.3.1

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} + sin (45) \cdot sin (30)

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

Step 3.3.2

Combine using the product rule for radicals.

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

Step 3.3.3

Multiply

$2$ by

$3$ .

$\frac{\sqrt{6}}{2 \cdot 2} + \sin (45) \cdot \sin (30)$

Step 3.3.4

Multiply

$2$ by

$2$ .

$\frac{\sqrt{6}}{4} + \sin (45) \cdot \sin (30)$

Step 3.4

The exact value of

$\sin (45)$ is

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

$\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \sin (30)$

Step 3.5

The exact value of

$\sin (30)$ is

$\frac{1}{2}$ .

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

Step 3.6

Multiply

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

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Step 3.6.1

Multiply

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

$\frac{1}{2}$ .

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

Step 3.6.2

Multiply

$2$ by

$2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.96592582 \dots$