Trigonometry Examples

$\cos (105) \cos (75)$

Step 1

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

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Step 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 second quadrant.

$- \cos (75) \cos (75)$

Step 1.2

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

$- \cos (30 + 45) \cos (75)$

Step 1.3

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

$- (\cos (30) \cos (45) - \sin (30) \sin (45)) \cos (75)$

Step 1.4

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

$- (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45)) \cos (75)$

Step 1.5

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

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)) \cos (75)$

Step 1.6

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

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)) \cos (75)$

Step 1.7

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

$- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (75)$

Step 1.8

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

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

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Step 1.8.1.1.1

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

$- (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (75)$

Step 1.8.1.1.2

Combine using the product rule for radicals.

$- (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (75)$

Step 1.8.1.1.3

Multiply $3$ by $2$ .

$- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (75)$

Step 1.8.1.1.4

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (75)$

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

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

Step 1.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.8.2

Combine the numerators over the common denominator.

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

Step 2

The exact value of $\cos (75)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1

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 2.2

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 2.3

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 2.4

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 2.5

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 2.6

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 2.7

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

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

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

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Step 2.7.1.1.1

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 2.7.1.1.2

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 2.7.1.1.3

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 2.7.1.1.4

Multiply $2$ by $2$ .

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

Step 2.7.1.2

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

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Step 2.7.1.2.1

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 2.7.1.2.2

Multiply $2$ by $2$ .

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

Step 2.7.2

Combine the numerators over the common denominator.

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

Step 3

Multiply $- \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 3.1

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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

Step 3.2

Raise $\sqrt{6} - \sqrt{2}$ to the power of $1$ .

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

Step 3.3

Raise $\sqrt{6} - \sqrt{2}$ to the power of $1$ .

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

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Multiply $4$ by $4$ .

$- \frac{{(\sqrt{6} - \sqrt{2})}^{2}}{16}$

Step 4

Rewrite ${(\sqrt{6} - \sqrt{2})}^{2}$ as $(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})$ .

$- \frac{(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})}{16}$

Step 5

Expand $(\sqrt{6} - \sqrt{2}) (\sqrt{6} - \sqrt{2})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$- \frac{\sqrt{6} (\sqrt{6} - \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2})}{16}$

Step 5.2

Apply the distributive property.

$- \frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} (\sqrt{6} - \sqrt{2})}{16}$

Step 5.3

Apply the distributive property.

$- \frac{\sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Combine using the product rule for radicals.

$- \frac{\sqrt{6 \cdot 6} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.2

Multiply $6$ by $6$ .

$- \frac{\sqrt{36} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.3

Rewrite $36$ as $6^{2}$ .

$- \frac{\sqrt{6^{2}} + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.4

Pull terms out from under the radical, assuming positive real numbers.

$- \frac{6 + \sqrt{6} (- \sqrt{2}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.5

Multiply $\sqrt{6} (- \sqrt{2})$ .

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Step 6.1.5.1

Combine using the product rule for radicals.

$- \frac{6 - \sqrt{6 \cdot 2} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.5.2

Multiply $6$ by $2$ .

$- \frac{6 - \sqrt{12} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.6

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.1.6.1

Factor $4$ out of $12$ .

$- \frac{6 - \sqrt{4 (3)} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.6.2

Rewrite $4$ as $2^{2}$ .

$- \frac{6 - \sqrt{2^{2} \cdot 3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.7

Pull terms out from under the radical.

$- \frac{6 - (2 \sqrt{3}) - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.8

Multiply $2$ by $- 1$ .

$- \frac{6 - 2 \sqrt{3} - \sqrt{2} \sqrt{6} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.9

Multiply $- \sqrt{2} \sqrt{6}$ .

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Step 6.1.9.1

Combine using the product rule for radicals.

$- \frac{6 - 2 \sqrt{3} - \sqrt{6 \cdot 2} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.9.2

Multiply $6$ by $2$ .

$- \frac{6 - 2 \sqrt{3} - \sqrt{12} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.10

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.1.10.1

Factor $4$ out of $12$ .

$- \frac{6 - 2 \sqrt{3} - \sqrt{4 (3)} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.10.2

Rewrite $4$ as $2^{2}$ .

$- \frac{6 - 2 \sqrt{3} - \sqrt{2^{2} \cdot 3} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.11

Pull terms out from under the radical.

$- \frac{6 - 2 \sqrt{3} - (2 \sqrt{3}) - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.12

Multiply $2$ by $- 1$ .

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{2} (- \sqrt{2})}{16}$

Step 6.1.13

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 6.1.13.1

Multiply $- 1$ by $- 1$ .

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{2} \sqrt{2}}{16}$

Step 6.1.13.2

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

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{2} \sqrt{2}}{16}$

Step 6.1.13.3

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} \sqrt{2}}{16}$

Step 6.1.13.4

Raise $\sqrt{2}$ to the power of $1$ .

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}}{16}$

Step 6.1.13.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{1 + 1}}{16}$

Step 6.1.13.6

Add $1$ and $1$ .

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{2}}^{2}}{16}$

Step 6.1.14

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 6.1.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + {(2^{\frac{1}{2}})}^{2}}{16}$

Step 6.1.14.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{1}{2} \cdot 2}}{16}$

Step 6.1.14.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}}}{16}$

Step 6.1.14.4

Cancel the common factor of $2$ .

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Step 6.1.14.4.1

Cancel the common factor.

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{\frac{2}{2}}}{16}$

Step 6.1.14.4.2

Rewrite the expression.

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2^{1}}{16}$

Step 6.1.14.5

Evaluate the exponent.

$- \frac{6 - 2 \sqrt{3} - 2 \sqrt{3} + 2}{16}$

Step 6.2

Add $6$ and $2$ .

$- \frac{8 - 2 \sqrt{3} - 2 \sqrt{3}}{16}$

Step 6.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$- \frac{8 - 4 \sqrt{3}}{16}$

Step 7

Cancel the common factor of $8 - 4 \sqrt{3}$ and $16$ .

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Step 7.1

Factor $4$ out of $8$ .

$- \frac{4 (2) - 4 \sqrt{3}}{16}$

Step 7.2

Factor $4$ out of $- 4 \sqrt{3}$ .

$- \frac{4 (2) + 4 (- \sqrt{3})}{16}$

Step 7.3

Factor $4$ out of $4 (2) + 4 (- \sqrt{3})$ .

$- \frac{4 (2 - \sqrt{3})}{16}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $4$ out of $16$ .

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

Step 7.4.2

Cancel the common factor.

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

Step 7.4.3

Rewrite the expression.

$- \frac{2 - \sqrt{3}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 - \sqrt{3}}{4}$

Decimal Form:

$- 0.06698729 \dots$