Trigonometry Examples

$\sin (105) \sin (75)$

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.

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

Step 1.2

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

$\sin (30 + 45) \sin (75)$

Step 1.3

Apply the sum of angles identity.

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

Step 1.4

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

$(\frac{1}{2} \cos (45) + \cos (30) \sin (45)) \sin (75)$

Step 1.5

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) \sin (75)$

Step 1.6

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

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

Step 1.7

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (75)$

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

$(\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (75)$

Step 1.8.1.1.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (75)$

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

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \sin (75)$

Step 1.8.1.2.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \sin (75)$

Step 1.8.1.2.3

Multiply $3$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) \sin (75)$

Step 1.8.1.2.4

Multiply $2$ by $2$ .

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

Step 1.8.2

Combine the numerators over the common denominator.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Apply the sum of angles identity.

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

Step 2.3

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{1}{2} \cos (45) + \cos (30) \sin (45))$

Step 2.4

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

$\frac{\sqrt{2} + \sqrt{6}}{4} (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45))$

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

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

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Step 2.7.1.1.1

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

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

Step 2.7.1.1.2

Multiply $2$ by $2$ .

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

Step 2.7.1.2

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

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Step 2.7.1.2.1

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

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

Step 2.7.1.2.2

Combine using the product rule for radicals.

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

Step 2.7.1.2.3

Multiply $3$ by $2$ .

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

Step 2.7.1.2.4

Multiply $2$ by $2$ .

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

Step 2.7.2

Combine the numerators over the common denominator.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Multiply $4$ by $4$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

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.

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

Step 6.1.2

Multiply $2$ by $2$ .

$\frac{\sqrt{4} + \sqrt{2} \sqrt{6} + \sqrt{6} \sqrt{2} + \sqrt{6} \sqrt{6}}{16}$

Step 6.1.3

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

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

Step 6.1.4

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

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

Step 6.1.5

Combine using the product rule for radicals.

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

Step 6.1.6

Multiply $2$ by $6$ .

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

Step 6.1.7

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

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Step 6.1.7.1

Factor $4$ out of $12$ .

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

Step 6.1.7.2

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

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

Step 6.1.8

Pull terms out from under the radical.

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

Step 6.1.9

Combine using the product rule for radicals.

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

Step 6.1.10

Multiply $6$ by $2$ .

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

Step 6.1.11

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

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Step 6.1.11.1

Factor $4$ out of $12$ .

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

Step 6.1.11.2

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

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

Step 6.1.12

Pull terms out from under the radical.

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

Step 6.1.13

Combine using the product rule for radicals.

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

Step 6.1.14

Multiply $6$ by $6$ .

$\frac{2 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{36}}{16}$

Step 6.1.15

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

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

Step 6.1.16

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

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

Step 6.2

Add $2$ and $6$ .

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

Step 6.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

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

Step 7

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

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Step 7.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 2 + 4 \sqrt{3}}{16}$

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $4$ out of $16$ .

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

Step 7.4.2

Cancel the common factor.

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

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:

$0.93301270 \dots$