Trigonometry Examples

$(1, 3)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(1, 3)$ , draw the triangle between the three points $(0, 0)$ , $(1, 0)$ , and $(1, 3)$ .

Opposite : $3$

Adjacent : $1$

Step 2

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

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Step 2.1

One to any power is one.

$\sqrt{1 + {(3)}^{2}}$

Step 2.2

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

$\sqrt{1 + 9}$

Step 2.3

Add $1$ and $9$ .

$\sqrt{10}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{1}{\sqrt{10}}$ .

$\frac{1}{\sqrt{10}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (θ) = \frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (θ) = \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.2.2

Raise $\sqrt{10}$ to the power of $1$ .

$\cos (θ) = \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.2.3

Raise $\sqrt{10}$ to the power of $1$ .

$\cos (θ) = \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 4.2.4

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

$\cos (θ) = \frac{\sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 4.2.5

Add $1$ and $1$ .

$\cos (θ) = \frac{\sqrt{10}}{{\sqrt{10}}^{2}}$

Step 4.2.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 4.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\cos (θ) = \frac{\sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

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

$\cos (θ) = \frac{\sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

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

$\cos (θ) = \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

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.

$\cos (θ) = \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{\sqrt{10}}{10}$

Step 4.2.6.5

Evaluate the exponent.

$\cos (θ) = \frac{\sqrt{10}}{10}$

Step 5

Approximate the result.

$\cos (θ) = \frac{\sqrt{10}}{10} \approx 0.31622776$

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