Trigonometry Examples

cos (60 ° - 45 °)

$\cos (60 ° - 45 °)$

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 (60 °) \cdot cos (45 °) + sin (60 °) \cdot sin (45 °)

$\cos (60 °) \cdot \cos (45 °) + \sin (60 °) \cdot \sin (45 °)$

Step 2

Remove parentheses.

cos (60 °) \cdot cos (45 °) + sin (60 °) \cdot sin (45 °)

$\cos (60 °) \cdot \cos (45 °) + \sin (60 °) \cdot \sin (45 °)$

Step 3

Simplify each term.

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Step 3.1

The exact value of

cos (60 °)

$\cos (60 °)$ is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{2} \cdot cos (45 °) + sin (60 °) \cdot sin (45 °)

$\frac{1}{2} \cdot \cos (45 °) + \sin (60 °) \cdot \sin (45 °)$

Step 3.2

The exact value of

cos (45 °)

$\cos (45 °)$ is

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

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

\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + sin (60 °) \cdot sin (45 °)

$\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \sin (60 °) \cdot \sin (45 °)$

Step 3.3

Multiply

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

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

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Step 3.3.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

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

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

\frac{\sqrt{2}}{2 \cdot 2} + sin (60 °) \cdot sin (45 °)

$\frac{\sqrt{2}}{2 \cdot 2} + \sin (60 °) \cdot \sin (45 °)$

Step 3.3.2

Multiply

2

$2$ by

2

$2$ .

\frac{\sqrt{2}}{4} + sin (60 °) \cdot sin (45 °)

$\frac{\sqrt{2}}{4} + \sin (60 °) \cdot \sin (45 °)$

\frac{\sqrt{2}}{4} + sin (60 °) \cdot sin (45 °)

$\frac{\sqrt{2}}{4} + \sin (60 °) \cdot \sin (45 °)$

Step 3.4

The exact value of

sin (60 °)

$\sin (60 °)$ is

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

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

\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot sin (45 °)

$\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \sin (45 °)$

Step 3.5

The exact value of

sin (45 °)

$\sin (45 °)$ is

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

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

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

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

Step 3.6

Multiply

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

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

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Step 3.6.1

Multiply

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

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

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

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

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

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

Step 3.6.2

Combine using the product rule for radicals.

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

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

Step 3.6.3

Multiply

3

$3$ by

2

$2$ .

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

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

Step 3.6.4

Multiply

2

$2$ by

2

$2$ .

\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}

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.96592582 \dots

$0.96592582 \dots$