Trigonometry Examples

$(- 2, 7)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 2, 7)$ , draw the triangle between the three points $(0, 0)$ , $(- 2, 0)$ , and $(- 2, 7)$ .

Opposite : $7$

Adjacent : $- 2$

Step 2

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

Tap for more steps...

Step 2.1

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

$\sqrt{4 + {(7)}^{2}}$

Step 2.2

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

$\sqrt{4 + 49}$

Step 2.3

Add $4$ and $49$ .

$\sqrt{53}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 2}{\sqrt{53}}$ .

$\frac{- 2}{\sqrt{53}}$

Step 4

Simplify $\cos (θ)$ .

Tap for more steps...

Step 4.1

Move the negative in front of the fraction.

$\cos (θ) = - \frac{2}{\sqrt{53}}$

Step 4.2

Multiply $\frac{2}{\sqrt{53}}$ by $\frac{\sqrt{53}}{\sqrt{53}}$ .

$\cos (θ) = - (\frac{2}{\sqrt{53}} \cdot \frac{\sqrt{53}}{\sqrt{53}})$

Step 4.3

Combine and simplify the denominator.

Tap for more steps...

Step 4.3.1

Multiply $\frac{2}{\sqrt{53}}$ by $\frac{\sqrt{53}}{\sqrt{53}}$ .

$\cos (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 4.3.2

Raise $\sqrt{53}$ to the power of $1$ .

$\cos (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 4.3.3

Raise $\sqrt{53}$ to the power of $1$ .

$\cos (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 4.3.4

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

$\cos (θ) = - \frac{2 \sqrt{53}}{{\sqrt{53}}^{1 + 1}}$

Step 4.3.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{2 \sqrt{53}}{{\sqrt{53}}^{2}}$

Step 4.3.6

Rewrite ${\sqrt{53}}^{2}$ as $53$ .

Tap for more steps...

Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{53}$ as $53^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{2 \sqrt{53}}{{(53^{\frac{1}{2}})}^{2}}$

Step 4.3.6.2

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

$\cos (θ) = - \frac{2 \sqrt{53}}{53^{\frac{1}{2} \cdot 2}}$

Step 4.3.6.3

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

$\cos (θ) = - \frac{2 \sqrt{53}}{53^{\frac{2}{2}}}$

Step 4.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{2 \sqrt{53}}{53^{\frac{2}{2}}}$

Step 4.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{2 \sqrt{53}}{53}$

Step 4.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{2 \sqrt{53}}{53}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{2 \sqrt{53}}{53} \approx - 0.27472112$