Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

Apply the sum of angles identity.

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

Step 1.1.4

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

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

Step 1.1.5

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 1.1.6

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 1.1.7

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 1.1.8

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

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

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

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Step 1.1.8.1.1.1

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 1.1.8.1.1.2

Multiply $2$ by $2$ .

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

Step 1.1.8.1.2

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

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Step 1.1.8.1.2.1

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 1.1.8.1.2.2

Combine using the product rule for radicals.

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

Step 1.1.8.1.2.3

Multiply $3$ by $2$ .

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

Step 1.1.8.1.2.4

Multiply $2$ by $2$ .

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

Step 1.1.8.2

Combine the numerators over the common denominator.

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

Step 1.2

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

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Step 1.2.1

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 1.2.2

Separate negation.

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

Step 1.2.3

Apply the difference of angles identity.

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

Step 1.2.4

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 1.2.5

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 1.2.6

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 1.2.7

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 1.2.8

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

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

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

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Step 1.2.8.1.1.1

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 1.2.8.1.1.2

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 1.2.8.1.1.3

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 1.2.8.1.1.4

Multiply $2$ by $2$ .

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

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

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 1.2.8.1.2.2

Multiply $2$ by $2$ .

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

Step 1.2.8.2

Combine the numerators over the common denominator.

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- - \sqrt{2}$ .

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2

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

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

Step 4

Simplify terms.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $4$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.70710678 \dots$