Trigonometry Examples

$(7, 5)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(7, 5)$ , draw the triangle between the three points $(0, 0)$ , $(7, 0)$ , and $(7, 5)$ .

Opposite : $5$

Adjacent : $7$

Step 2

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

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Step 2.1

Raise $7$ to the power of $2$ .

$\sqrt{49 + {(5)}^{2}}$

Step 2.2

Raise $5$ to the power of $2$ .

$\sqrt{49 + 25}$

Step 2.3

Add $49$ and $25$ .

$\sqrt{74}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{7}{\sqrt{74}}$ .

$\frac{7}{\sqrt{74}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply $\frac{7}{\sqrt{74}}$ by $\frac{\sqrt{74}}{\sqrt{74}}$ .

$\cos (θ) = \frac{7}{\sqrt{74}} \cdot \frac{\sqrt{74}}{\sqrt{74}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{7}{\sqrt{74}}$ by $\frac{\sqrt{74}}{\sqrt{74}}$ .

$\cos (θ) = \frac{7 \sqrt{74}}{\sqrt{74} \sqrt{74}}$

Step 4.2.2

Raise $\sqrt{74}$ to the power of $1$ .

$\cos (θ) = \frac{7 \sqrt{74}}{\sqrt{74} \sqrt{74}}$

Step 4.2.3

Raise $\sqrt{74}$ to the power of $1$ .

$\cos (θ) = \frac{7 \sqrt{74}}{\sqrt{74} \sqrt{74}}$

Step 4.2.4

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

$\cos (θ) = \frac{7 \sqrt{74}}{{\sqrt{74}}^{1 + 1}}$

Step 4.2.5

Add $1$ and $1$ .

$\cos (θ) = \frac{7 \sqrt{74}}{{\sqrt{74}}^{2}}$

Step 4.2.6

Rewrite ${\sqrt{74}}^{2}$ as $74$ .

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Step 4.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{74}$ as $74^{\frac{1}{2}}$ .

$\cos (θ) = \frac{7 \sqrt{74}}{{(74^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

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

$\cos (θ) = \frac{7 \sqrt{74}}{74^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

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

$\cos (θ) = \frac{7 \sqrt{74}}{74^{\frac{2}{2}}}$

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.

$\cos (θ) = \frac{7 \sqrt{74}}{74^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{7 \sqrt{74}}{74}$

Step 4.2.6.5

Evaluate the exponent.

$\cos (θ) = \frac{7 \sqrt{74}}{74}$

Step 5

Approximate the result.

$\cos (θ) = \frac{7 \sqrt{74}}{74} \approx 0.81373347$