Trigonometry Examples

$\cos (195) + \cos (105)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (195)$ is $- \frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 1.1.1

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 third quadrant.

$- \cos (15) + \cos (105)$

Step 1.1.2

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

$- \cos (45 - 30) + \cos (105)$

Step 1.1.3

Separate negation.

$- \cos (45 - (30)) + \cos (105)$

Step 1.1.4

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$- (\cos (45) \cos (30) + \sin (45) \sin (30)) + \cos (105)$

Step 1.1.5

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) + \cos (105)$

Step 1.1.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) + \cos (105)$

Step 1.1.7

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) + \cos (105)$

Step 1.1.8

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105)$

Step 1.1.9

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

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Step 1.1.9.1

Simplify each term.

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Step 1.1.9.1.1

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

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Step 1.1.9.1.1.1

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

$- (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105)$

Step 1.1.9.1.1.2

Combine using the product rule for radicals.

$- (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105)$

Step 1.1.9.1.1.3

Multiply $2$ by $3$ .

$- (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105)$

Step 1.1.9.1.1.4

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + \cos (105)$

Step 1.1.9.1.2

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

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Step 1.1.9.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) + \cos (105)$

Step 1.1.9.1.2.2

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) + \cos (105)$

Step 1.1.9.2

Combine the numerators over the common denominator.

$- \frac{\sqrt{6} + \sqrt{2}}{4} + \cos (105)$

Step 1.2

The exact value of $\cos (105)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.2.1

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.

$- \frac{\sqrt{6} + \sqrt{2}}{4} - \cos (75)$

Step 1.2.2

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

$- \frac{\sqrt{6} + \sqrt{2}}{4} - \cos (30 + 45)$

Step 1.2.3

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$- \frac{\sqrt{6} + \sqrt{2}}{4} - (\cos (30) \cos (45) - \sin (30) \sin (45))$

Step 1.2.4

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{6} + \sqrt{2}}{4} - (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45))$

Step 1.2.5

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{6} + \sqrt{2}}{4} - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45))$

Step 1.2.6

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$- \frac{\sqrt{6} + \sqrt{2}}{4} - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45))$

Step 1.2.7

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{6} + \sqrt{2}}{4} - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 1.2.8

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

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

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

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Step 1.2.8.1.1.1

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

$- \frac{\sqrt{6} + \sqrt{2}}{4} - (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 1.2.8.1.1.2

Combine using the product rule for radicals.

$- \frac{\sqrt{6} + \sqrt{2}}{4} - (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$

Step 1.2.8.1.1.3

Multiply $3$ by $2$ .

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

Step 1.2.8.1.1.4

Multiply $2$ by $2$ .

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

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

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

Step 1.2.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.2.8.2

Combine the numerators over the common denominator.

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Multiply $- - \sqrt{2}$ .

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2

Multiply $\sqrt{2}$ by $1$ .

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

Step 4

Simplify terms.

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Step 4.1

Subtract $\sqrt{6}$ from $- \sqrt{6}$ .

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

Step 4.2

Add $- \sqrt{2}$ and $\sqrt{2}$ .

$\frac{- 2 \sqrt{6} + 0}{4}$

Step 4.3

Add $- 2 \sqrt{6}$ and $0$ .

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

Step 4.4

Cancel the common factor of $- 2$ and $4$ .

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Step 4.4.1

Factor $2$ out of $- 2 \sqrt{6}$ .

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $4$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 4.5

Move the negative in front of the fraction.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.22474487 \dots$