Trigonometry Examples

$(8, 15 °)$

Step 1

Use the conversion formulas to convert from polar coordinates to rectangular coordinates.

$x = r c o s θ$

$y = r s i n θ$

Step 2

Substitute in the known values of $r = 8$ and $θ = 15 °$ into the formulas.

$x = (8) \cos (15 °)$

$y = (8) \sin (15 °)$

Step 3

The exact value of $\cos (15 °)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 3.1

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

$x = 8 \cos (45 - 30)$

$y = (8) \sin (15 °)$

Step 3.2

Separate negation.

$x = 8 \cos (45 - (30))$

$y = (8) \sin (15 °)$

Step 3.3

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

$x = 8 (\cos (45) \cos (30) + \sin (45) \sin (30))$

$y = (8) \sin (15 °)$

Step 3.4

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

$x = 8 (\frac{\sqrt{2}}{2} \cdot \cos (30) + \sin (45) \sin (30))$

$y = (8) \sin (15 °)$

Step 3.5

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

$x = 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30))$

$y = (8) \sin (15 °)$

Step 3.6

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

$x = 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \sin (30))$

$y = (8) \sin (15 °)$

Step 3.7

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

$x = 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (8) \sin (15 °)$

Step 3.8

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

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Step 3.8.1

Simplify each term.

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Step 3.8.1.1

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

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Step 3.8.1.1.1

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

$x = 8 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (8) \sin (15 °)$

Step 3.8.1.1.2

Combine using the product rule for radicals.

$x = 8 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (8) \sin (15 °)$

Step 3.8.1.1.3

Multiply $2$ by $3$ .

$x = 8 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (8) \sin (15 °)$

Step 3.8.1.1.4

Multiply $2$ by $2$ .

$x = 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (8) \sin (15 °)$

$x = 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$y = (8) \sin (15 °)$

Step 3.8.1.2

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

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Step 3.8.1.2.1

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

$x = 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2})$

$y = (8) \sin (15 °)$

Step 3.8.1.2.2

Multiply $2$ by $2$ .

$x = 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

$y = (8) \sin (15 °)$

$x = 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

$y = (8) \sin (15 °)$

$x = 8 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

$y = (8) \sin (15 °)$

Step 3.8.2

Combine the numerators over the common denominator.

$x = 8 (\frac{\sqrt{6} + \sqrt{2}}{4})$

$y = (8) \sin (15 °)$

$x = 8 (\frac{\sqrt{6} + \sqrt{2}}{4})$

$y = (8) \sin (15 °)$

$x = 8 (\frac{\sqrt{6} + \sqrt{2}}{4})$

$y = (8) \sin (15 °)$

Step 4

Cancel the common factor of $4$ .

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Step 4.1

Factor $4$ out of $8$ .

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

$y = (8) \sin (15 °)$

Step 4.2

Cancel the common factor.

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

$y = (8) \sin (15 °)$

Step 4.3

Rewrite the expression.

$x = 2 (\sqrt{6} + \sqrt{2})$

$y = (8) \sin (15 °)$

$x = 2 (\sqrt{6} + \sqrt{2})$

$y = (8) \sin (15 °)$

Step 5

Apply the distributive property.

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = (8) \sin (15 °)$

Step 6

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

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Step 6.1

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

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 \sin (45 - 30)$

Step 6.2

Separate negation.

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 \sin (45 - (30))$

Step 6.3

Apply the difference of angles identity.

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\sin (45) \cos (30) - \cos (45) \sin (30))$

Step 6.4

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

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{2}}{2} \cdot \cos (30) - \cos (45) \sin (30))$

Step 6.5

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

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

Step 6.6

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

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \sin (30))$

Step 6.7

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

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 6.8

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

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Step 6.8.1

Simplify each term.

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Step 6.8.1.1

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

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Step 6.8.1.1.1

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

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 6.8.1.1.2

Combine using the product rule for radicals.

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 6.8.1.1.3

Multiply $2$ by $3$ .

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 6.8.1.1.4

Multiply $2$ by $2$ .

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 6.8.1.2

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

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Step 6.8.1.2.1

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

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Step 6.8.1.2.2

Multiply $2$ by $2$ .

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Step 6.8.2

Combine the numerators over the common denominator.

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6} - \sqrt{2}}{4})$

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6} - \sqrt{2}}{4})$

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 8 (\frac{\sqrt{6} - \sqrt{2}}{4})$

Step 7

Cancel the common factor of $4$ .

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Step 7.1

Factor $4$ out of $8$ .

$x = 2 \sqrt{6} + 2 \sqrt{2}$

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

Step 7.2

Cancel the common factor.

$x = 2 \sqrt{6} + 2 \sqrt{2}$

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

Step 7.3

Rewrite the expression.

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 2 (\sqrt{6} - \sqrt{2})$

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 2 (\sqrt{6} - \sqrt{2})$

Step 8

Apply the distributive property.

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 2 \sqrt{6} + 2 (- \sqrt{2})$

Step 9

Multiply $- 1$ by $2$ .

$x = 2 \sqrt{6} + 2 \sqrt{2}$

$y = 2 \sqrt{6} - 2 \sqrt{2}$

Step 10

The rectangular representation of the polar point $(8, 15 °)$ is $(2 \sqrt{6} + 2 \sqrt{2}, 2 \sqrt{6} - 2 \sqrt{2})$ .

$(2 \sqrt{6} + 2 \sqrt{2}, 2 \sqrt{6} - 2 \sqrt{2})$