Trigonometry Examples

$\sin (x) = \frac{8}{9}$ , $\cos (2 x)$

Step 1

Use the definition of sine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\sin (x) = \frac{opposite}{hypotenuse}$

Step 2

Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.

$Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = \sqrt{{(9)}^{2} - {(8)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise $9$ to the power of $2$ .

Adjacent $= \sqrt{81 - {(8)}^{2}}$

Step 4.2

Raise $8$ to the power of $2$ .

Adjacent $= \sqrt{81 - 1 \cdot 64}$

Step 4.3

Multiply $- 1$ by $64$ .

Adjacent $= \sqrt{81 - 64}$

Step 4.4

Subtract $64$ from $81$ .

Adjacent $= \sqrt{17}$

Step 5

Use the definition of cosine to find the value of $\cos (x)$ .

$\cos (x) = \frac{adjacent}{hypotenuse}$

Step 6

Substitute in the known values.

$\cos (x) = \frac{\sqrt{17}}{9}$

Step 7

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

$2 \cos^{2} (x) - 1$

Step 8

Use the definition of $c o s$ to find the value of $\cos (x)$ . In this case, $\cos (x) = \frac{\sqrt{17}}{9}$ .

$\cos (x) = \frac{\sqrt{17}}{9}$

Step 9

Substitute the values into $2 \cos^{2} (x) - 1$ .

$\cos (2 x) = 2 {(\frac{\sqrt{17}}{9})}^{2} - 1$

Step 10

Evaluate $2 {(\frac{\sqrt{17}}{9})}^{2} - 1$ to find $\cos (2 x)$ .

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Step 10.1

Simplify each term.

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Step 10.1.1

Apply the product rule to $\frac{\sqrt{17}}{9}$ .

$\cos (2 x) = 2 (\frac{{\sqrt{17}}^{2}}{9^{2}}) - 1$

Step 10.1.2

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 10.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\cos (2 x) = 2 (\frac{{(17^{\frac{1}{2}})}^{2}}{9^{2}}) - 1$

Step 10.1.2.2

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

$\cos (2 x) = 2 (\frac{17^{\frac{1}{2} \cdot 2}}{9^{2}}) - 1$

Step 10.1.2.3

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

$\cos (2 x) = 2 (\frac{17^{\frac{2}{2}}}{9^{2}}) - 1$

Step 10.1.2.4

Cancel the common factor of $2$ .

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Step 10.1.2.4.1

Cancel the common factor.

$\cos (2 x) = 2 (\frac{17^{\frac{2}{2}}}{9^{2}}) - 1$

Step 10.1.2.4.2

Rewrite the expression.

$\cos (2 x) = 2 (\frac{17}{9^{2}}) - 1$

Step 10.1.2.5

Evaluate the exponent.

$\cos (2 x) = 2 (\frac{17}{9^{2}}) - 1$

Step 10.1.3

Raise $9$ to the power of $2$ .

$\cos (2 x) = 2 (\frac{17}{81}) - 1$

Step 10.1.4

Multiply $2 (\frac{17}{81})$ .

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Step 10.1.4.1

Combine $2$ and $\frac{17}{81}$ .

$\cos (2 x) = \frac{2 \cdot 17}{81} - 1$

Step 10.1.4.2

Multiply $2$ by $17$ .

$\cos (2 x) = \frac{34}{81} - 1$

Step 10.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$\cos (2 x) = \frac{34}{81} - 1 \cdot \frac{81}{81}$

Step 10.3

Combine $- 1$ and $\frac{81}{81}$ .

$\cos (2 x) = \frac{34}{81} + \frac{- 1 \cdot 81}{81}$

Step 10.4

Combine the numerators over the common denominator.

$\cos (2 x) = \frac{34 - 1 \cdot 81}{81}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Multiply $- 1$ by $81$ .

$\cos (2 x) = \frac{34 - 81}{81}$

Step 10.5.2

Subtract $81$ from $34$ .

$\cos (2 x) = \frac{- 47}{81}$

Step 10.6

Move the negative in front of the fraction.

$\cos (2 x) = - \frac{47}{81}$