Trigonometry Examples

$(22, 14)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(22, 14)$ , draw the triangle between the three points $(0, 0)$ , $(22, 0)$ , and $(22, 14)$ .

Opposite : $14$

Adjacent : $22$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise $22$ to the power of $2$ .

$\sqrt{484 + {(14)}^{2}}$

Step 2.2

Raise $14$ to the power of $2$ .

$\sqrt{484 + 196}$

Step 2.3

Add $484$ and $196$ .

$\sqrt{680}$

Step 2.4

Rewrite $680$ as $2^{2} \cdot 170$ .

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Step 2.4.1

Factor $4$ out of $680$ .

$\sqrt{4 (170)}$

Step 2.4.2

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

$\sqrt{2^{2} \cdot 170}$

Step 2.5

Pull terms out from under the radical.

$2 \sqrt{170}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{22}{2 \sqrt{170}}$ .

$\frac{22}{2 \sqrt{170}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Cancel the common factor of $22$ and $2$ .

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Step 4.1.1

Factor $2$ out of $22$ .

$\cos (θ) = \frac{2 \cdot 11}{2 \sqrt{170}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $2 \sqrt{170}$ .

$\cos (θ) = \frac{2 \cdot 11}{2 (\sqrt{170})}$

Step 4.1.2.2

Cancel the common factor.

$\cos (θ) = \frac{2 \cdot 11}{2 \sqrt{170}}$

Step 4.1.2.3

Rewrite the expression.

$\cos (θ) = \frac{11}{\sqrt{170}}$

Step 4.2

Multiply $\frac{11}{\sqrt{170}}$ by $\frac{\sqrt{170}}{\sqrt{170}}$ .

$\cos (θ) = \frac{11}{\sqrt{170}} \cdot \frac{\sqrt{170}}{\sqrt{170}}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{11}{\sqrt{170}}$ by $\frac{\sqrt{170}}{\sqrt{170}}$ .

$\cos (θ) = \frac{11 \sqrt{170}}{\sqrt{170} \sqrt{170}}$

Step 4.3.2

Raise $\sqrt{170}$ to the power of $1$ .

$\cos (θ) = \frac{11 \sqrt{170}}{\sqrt{170} \sqrt{170}}$

Step 4.3.3

Raise $\sqrt{170}$ to the power of $1$ .

$\cos (θ) = \frac{11 \sqrt{170}}{\sqrt{170} \sqrt{170}}$

Step 4.3.4

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

$\cos (θ) = \frac{11 \sqrt{170}}{{\sqrt{170}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\cos (θ) = \frac{11 \sqrt{170}}{{\sqrt{170}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{170}}^{2}$ as $170$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{170}$ as $170^{\frac{1}{2}}$ .

$\cos (θ) = \frac{11 \sqrt{170}}{{(170^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\cos (θ) = \frac{11 \sqrt{170}}{170^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\cos (θ) = \frac{11 \sqrt{170}}{170^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\cos (θ) = \frac{11 \sqrt{170}}{170^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{11 \sqrt{170}}{170}$

Step 4.3.6.5

Evaluate the exponent.

$\cos (θ) = \frac{11 \sqrt{170}}{170}$

Step 5

Approximate the result.

$\cos (θ) = \frac{11 \sqrt{170}}{170} \approx 0.84366148$