Trigonometry Examples

sin (195)

$\sin (195)$

Step 1

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 third quadrant.

- sin (15)

$- \sin (15)$

Step 2

Split

15

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

- sin (45 - 30)

$- \sin (45 - 30)$

Step 3

Separate negation.

- sin (45 - (30))

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

Step 4

Apply the difference of angles identity.

- (sin (45) cos (30) - cos (45) sin (30))

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

Step 5

The exact value of

sin (45)

$\sin (45)$ is

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

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

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

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

Step 6

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} - cos (45) sin (30))

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

Step 7

The exact value of

cos (45)

$\cos (45)$ 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 8

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 9

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 9.1

Simplify each term.

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Step 9.1.1

Multiply

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

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

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Step 9.1.1.1

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 9.1.1.2

Combine using the product rule for radicals.

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

Step 9.1.1.3

Multiply

$2$ by

$3$ .

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

Step 9.1.1.4

Multiply

$2$ by

$2$ .

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

Step 9.1.2

Multiply

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

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Step 9.1.2.1

Multiply

$\frac{1}{2}$ by

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

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

Step 9.1.2.2

Multiply

$2$ by

$2$ .

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

Step 9.2

Combine the numerators over the common denominator.

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25881904 \dots$