Trigonometry Examples

$(45, 24)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(45, 24)$ , draw the triangle between the three points $(0, 0)$ , $(45, 0)$ , and $(45, 24)$ .

Opposite : $24$

Adjacent : $45$

Step 2

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

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Step 2.1

Raise $45$ to the power of $2$ .

$\sqrt{2025 + {(24)}^{2}}$

Step 2.2

Raise $24$ to the power of $2$ .

$\sqrt{2025 + 576}$

Step 2.3

Add $2025$ and $576$ .

$\sqrt{2601}$

Step 2.4

Rewrite $2601$ as $51^{2}$ .

$\sqrt{51^{2}}$

Step 2.5

Pull terms out from under the radical, assuming positive real numbers.

$51$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{45}{51}$ .

$\frac{45}{51}$

Step 4

Cancel the common factor of $45$ and $51$ .

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Step 4.1

Factor $3$ out of $45$ .

$\cos (θ) = \frac{3 (15)}{51}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $51$ .

$\cos (θ) = \frac{3 \cdot 15}{3 \cdot 17}$

Step 4.2.2

Cancel the common factor.

$\cos (θ) = \frac{3 \cdot 15}{3 \cdot 17}$

Step 4.2.3

Rewrite the expression.

$\cos (θ) = \frac{15}{17}$

Step 5

Approximate the result.

$\cos (θ) = \frac{15}{17} \approx 0.88235294$