Trigonometry Examples Right Triangle Trigonometry

Find the Cosine Given the Point

$(3, 8)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(3, 8)$ , draw the triangle between the three points $(0, 0)$ , $(3, 0)$ , and $(3, 8)$ .

Opposite : $8$

Adjacent : $3$

Step 2

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

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Step 2.1

Raise $3$ to the power of $2$ .

$\sqrt{9 + {(8)}^{2}}$

Step 2.2

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

$\sqrt{9 + 64}$

Step 2.3

Add $9$ and $64$ .

$\sqrt{73}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{3}{\sqrt{73}}$ .

$\frac{3}{\sqrt{73}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply $\frac{3}{\sqrt{73}}$ by $\frac{\sqrt{73}}{\sqrt{73}}$ .

$\cos (θ) = \frac{3}{\sqrt{73}} \cdot \frac{\sqrt{73}}{\sqrt{73}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{3}{\sqrt{73}}$ by $\frac{\sqrt{73}}{\sqrt{73}}$ .

$\cos (θ) = \frac{3 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

Step 4.2.2

Raise $\sqrt{73}$ to the power of $1$ .

$\cos (θ) = \frac{3 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

Step 4.2.3

Raise $\sqrt{73}$ to the power of $1$ .

$\cos (θ) = \frac{3 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

Step 4.2.4

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

$\cos (θ) = \frac{3 \sqrt{73}}{{\sqrt{73}}^{1 + 1}}$

Step 4.2.5

Add $1$ and $1$ .

$\cos (θ) = \frac{3 \sqrt{73}}{{\sqrt{73}}^{2}}$

Step 4.2.6

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 4.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$\cos (θ) = \frac{3 \sqrt{73}}{{(73^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

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

$\cos (θ) = \frac{3 \sqrt{73}}{73^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

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

$\cos (θ) = \frac{3 \sqrt{73}}{73^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$\cos (θ) = \frac{3 \sqrt{73}}{73^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{3 \sqrt{73}}{73}$

Step 4.2.6.5

Evaluate the exponent.

$\cos (θ) = \frac{3 \sqrt{73}}{73}$

Step 5

Approximate the result.

$\cos (θ) = \frac{3 \sqrt{73}}{73} \approx 0.35112344$

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Find the Cosine Given the Point

Right Triangle Trigonometry

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