Trigonometry Examples

\cos (345)

$\cos (345)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case,

345

$345$ can be split into

300 + 45

$300 + 45$ .

\cos (300 + 45)

$\cos (300 + 45)$

Step 2

Use the sum 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 (300) - \sin (45) \cdot \sin (300)

$\cos (45) \cdot \cos (300) - \sin (45) \cdot \sin (300)$

Step 3

Remove parentheses.

\cos (45) \cdot \cos (300) - \sin (45) \cdot \sin (300)

$\cos (45) \cdot \cos (300) - \sin (45) \cdot \sin (300)$

Step 4

Simplify each term.

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Step 4.1

The exact value of

\cos (45)

$\cos (45)$ is

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

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

\frac{\sqrt{2}}{2} \cdot \cos (300) - \sin (45) \cdot \sin (300)

$\frac{\sqrt{2}}{2} \cdot \cos (300) - \sin (45) \cdot \sin (300)$

Step 4.2

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

\frac{\sqrt{2}}{2} \cdot \cos (60) - \sin (45) \cdot \sin (300)

$\frac{\sqrt{2}}{2} \cdot \cos (60) - \sin (45) \cdot \sin (300)$

Step 4.3

The exact value of

\cos (60)

$\cos (60)$ is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{\sqrt{2}}{2} \cdot \frac{1}{2} - \sin (45) \cdot \sin (300)

$\frac{\sqrt{2}}{2} \cdot \frac{1}{2} - \sin (45) \cdot \sin (300)$

Step 4.4

Multiply

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

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

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Step 4.4.1

Multiply

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

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

\frac{1}{2}

$\frac{1}{2}$ .

\frac{\sqrt{2}}{2 \cdot 2} - \sin (45) \cdot \sin (300)

$\frac{\sqrt{2}}{2 \cdot 2} - \sin (45) \cdot \sin (300)$

Step 4.4.2

Multiply

2

$2$ by

2

$2$ .

\frac{\sqrt{2}}{4} - \sin (45) \cdot \sin (300)

$\frac{\sqrt{2}}{4} - \sin (45) \cdot \sin (300)$

\frac{\sqrt{2}}{4} - \sin (45) \cdot \sin (300)

$\frac{\sqrt{2}}{4} - \sin (45) \cdot \sin (300)$

Step 4.5

The exact value of

\sin (45)

$\sin (45)$ is

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

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

\frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \sin (300)

$\frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot \sin (300)$

Step 4.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

\frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot (- \sin (60))

$\frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot (- \sin (60))$

Step 4.7

The exact value of

\sin (60)

$\sin (60)$ is

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

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

\frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot (- \frac{\sqrt{3}}{2})

$\frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{2} \cdot (- \frac{\sqrt{3}}{2})$

Step 4.8

Multiply

- \frac{\sqrt{2}}{2} (- \frac{\sqrt{3}}{2})

$- \frac{\sqrt{2}}{2} (- \frac{\sqrt{3}}{2})$ .

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Step 4.8.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{\sqrt{2}}{4} + 1 \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2}

$\frac{\sqrt{2}}{4} + 1 \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2}$

Step 4.8.2

Multiply

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

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

1

$1$ .

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

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

Step 4.8.3

Multiply

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

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

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

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

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

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

Step 4.8.4

Combine using the product rule for radicals.

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

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

Step 4.8.5

Multiply

2

$2$ by

3

$3$ .

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

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

Step 4.8.6

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 5

Combine the numerators over the common denominator.

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

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

Step 6

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

$0.96592582 \dots$