Trigonometry Examples

$\sin (105) \sin (15)$

Step 1

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

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Step 1.1

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

$\sin (75) \sin (15)$

Step 1.2

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

$\sin (30 + 45) \sin (15)$

Step 1.3

Apply the sum of angles identity.

$(\sin (30) \cos (45) + \cos (30) \sin (45)) \sin (15)$

Step 1.4

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

$(\frac{1}{2} \cos (45) + \cos (30) \sin (45)) \sin (15)$

Step 1.5

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) \sin (15)$

Step 1.6

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)) \sin (15)$

Step 1.7

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (15)$

Step 1.8

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

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

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Step 1.8.1.1.1

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

$(\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (15)$

Step 1.8.1.1.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (15)$

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \sin (15)$

Step 1.8.1.2.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \sin (15)$

Step 1.8.1.2.3

Multiply $3$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) \sin (15)$

Step 1.8.1.2.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}) \sin (15)$

Step 1.8.2

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} \sin (15)$

Step 2

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1

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

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

Step 2.2

Separate negation.

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

Step 2.3

Apply the difference of angles identity.

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

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

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Step 2.8.1.1.1

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

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

Step 2.8.1.1.2

Combine using the product rule for radicals.

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

Step 2.8.1.1.3

Multiply $2$ by $3$ .

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

Step 2.8.1.1.4

Multiply $2$ by $2$ .

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

Step 2.8.1.2

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

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Step 2.8.1.2.1

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

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

Step 2.8.1.2.2

Multiply $2$ by $2$ .

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

Step 2.8.2

Combine the numerators over the common denominator.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $4$ by $4$ .

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Combine using the product rule for radicals.

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

Step 4.2.1.2

Multiply $2$ by $6$ .

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

Factor $4$ out of $12$ .

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

Step 4.2.1.3.2

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

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

Step 4.2.1.4

Pull terms out from under the radical.

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

Step 4.2.1.5

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

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Step 4.2.1.5.1

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

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

Step 4.2.1.5.2

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

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

Step 4.2.1.5.3

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

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

Step 4.2.1.5.4

Add $1$ and $1$ .

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

Step 4.2.1.6

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

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Step 4.2.1.6.1

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

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

Step 4.2.1.6.2

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

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

Step 4.2.1.6.3

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

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

Step 4.2.1.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.6.4.1

Cancel the common factor.

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

Step 4.2.1.6.4.2

Rewrite the expression.

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

Step 4.2.1.6.5

Evaluate the exponent.

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

Step 4.2.1.7

Multiply $- 1$ by $2$ .

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

Step 4.2.1.8

Combine using the product rule for radicals.

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

Step 4.2.1.9

Multiply $6$ by $6$ .

$\frac{2 \sqrt{3} - 2 + \sqrt{36} + \sqrt{6} (- \sqrt{2})}{16}$

Step 4.2.1.10

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

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

Step 4.2.1.11

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

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

Step 4.2.1.12

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

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Step 4.2.1.12.1

Combine using the product rule for radicals.

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

Step 4.2.1.12.2

Multiply $6$ by $2$ .

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

Step 4.2.1.13

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

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Step 4.2.1.13.1

Factor $4$ out of $12$ .

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

Step 4.2.1.13.2

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

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

Step 4.2.1.14

Pull terms out from under the radical.

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

Step 4.2.1.15

Multiply $2$ by $- 1$ .

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

Step 4.2.2

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

$\frac{0 - 2 + 6}{16}$

Step 4.2.3

Subtract $2$ from $0$ .

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

Step 4.2.4

Add $- 2$ and $6$ .

$\frac{4}{16}$

Step 5

Cancel the common factor of $4$ and $16$ .

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Step 5.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{16}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $4$ out of $16$ .

$\frac{4 \cdot 1}{4 \cdot 4}$

Step 5.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 4}$

Step 5.2.3

Rewrite the expression.

$\frac{1}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$