Trigonometry Examples

$(- 10, 10 \sqrt{3})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 10, 10 \sqrt{3})$ , draw the triangle between the three points $(0, 0)$ , $(- 10, 0)$ , and $(- 10, 10 \sqrt{3})$ .

Opposite : $10 \sqrt{3}$

Adjacent : $- 10$

Step 2

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

Tap for more steps...

Step 2.1

Simplify the expression.

Tap for more steps...

Step 2.1.1

Raise $- 10$ to the power of $2$ .

$\sqrt{100 + {(10 \sqrt{3})}^{2}}$

Step 2.1.2

Apply the product rule to $10 \sqrt{3}$ .

$\sqrt{100 + 10^{2} {\sqrt{3}}^{2}}$

Step 2.1.3

Raise $10$ to the power of $2$ .

$\sqrt{100 + 100 {\sqrt{3}}^{2}}$

Step 2.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{100 + 100 {(3^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{100 + 100 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{100 + 100 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.4.1

Cancel the common factor.

$\sqrt{100 + 100 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{100 + 100 \cdot 3^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{100 + 100 \cdot 3}$

Step 2.3

Simplify the expression.

Tap for more steps...

Step 2.3.1

Multiply $100$ by $3$ .

$\sqrt{100 + 300}$

Step 2.3.2

Add $100$ and $300$ .

$\sqrt{400}$

Step 2.3.3

Rewrite $400$ as $20^{2}$ .

$\sqrt{20^{2}}$

Step 2.3.4

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

$20$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 10}{20}$ .

$\frac{- 10}{20}$

Step 4

Simplify $\cos (θ)$ .

Tap for more steps...

Step 4.1

Cancel the common factor of $- 10$ and $20$ .

Tap for more steps...

Step 4.1.1

Factor $10$ out of $- 10$ .

$\cos (θ) = \frac{10 (- 1)}{20}$

Step 4.1.2

Cancel the common factors.

Tap for more steps...

Step 4.1.2.1

Factor $10$ out of $20$ .

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

Step 4.1.2.2

Cancel the common factor.

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

Step 4.1.2.3

Rewrite the expression.

$\cos (θ) = \frac{- 1}{2}$

Step 4.2

Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{2}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{1}{2} \approx - 0.5$