Trigonometry Examples

cos (105)

$\cos (105)$

Step 1

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

- cos (75)

$- \cos (75)$

Step 2

Split

75

$75$ into two angles where the values of the six trigonometric functions are known.

- cos (30 + 45)

$- \cos (30 + 45)$

Step 3

Apply the sum 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 (30) cos (45) - sin (30) sin (45))

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

Step 4

The exact value of

cos (30)

$\cos (30)$ is

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

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

- (\frac{\sqrt{3}}{2} cos (45) - sin (30) sin (45))

$- (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45))$

Step 5

The exact value of

cos (45)

$\cos (45)$ is

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

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

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

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

Step 6

The exact value of

sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} sin (45))

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45))$

Step 7

The exact value of

sin (45)

$\sin (45)$ is

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

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

- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 8

Simplify

- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply

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

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

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Step 8.1.1.1

Multiply

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

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

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

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

- (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})

$- (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 8.1.1.2

Combine using the product rule for radicals.

- (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})

$- (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 8.1.1.3

Multiply

3

$3$ by

2

$2$ .

- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})

$- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 8.1.1.4

Multiply

2

$2$ by

2

$2$ .

- (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})

$- (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

- (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})

$- (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 8.1.2

Multiply

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

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

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Step 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 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 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}

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

- 0.25881904 \dots

$- 0.25881904 \dots$