Precalculus Examples

$(\frac{3 π}{2}, \frac{7 π}{4})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{3 π}{2}, \frac{7 π}{4})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{3 π}{2}, 0)$ , and $(\frac{3 π}{2}, \frac{7 π}{4})$ .

Opposite : $\frac{7 π}{4}$

Adjacent : $\frac{3 π}{2}$

Step 2

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

Tap for more steps...

Step 2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 2.1.1

Apply the product rule to $\frac{3 π}{2}$ .

$\sqrt{\frac{{(3 π)}^{2}}{2^{2}} + {(\frac{7 π}{4})}^{2}}$

Step 2.1.2

Apply the product rule to $3 π$ .

$\sqrt{\frac{3^{2} π^{2}}{2^{2}} + {(\frac{7 π}{4})}^{2}}$

Step 2.2

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

$\sqrt{\frac{9 π^{2}}{2^{2}} + {(\frac{7 π}{4})}^{2}}$

Step 2.3

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

$\sqrt{\frac{9 π^{2}}{4} + {(\frac{7 π}{4})}^{2}}$

Step 2.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 2.4.1

Apply the product rule to $\frac{7 π}{4}$ .

$\sqrt{\frac{9 π^{2}}{4} + \frac{{(7 π)}^{2}}{4^{2}}}$

Step 2.4.2

Apply the product rule to $7 π$ .

$\sqrt{\frac{9 π^{2}}{4} + \frac{7^{2} π^{2}}{4^{2}}}$

Step 2.5

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

$\sqrt{\frac{9 π^{2}}{4} + \frac{49 π^{2}}{4^{2}}}$

Step 2.6

Raise $4$ to the power of $2$ .

$\sqrt{\frac{9 π^{2}}{4} + \frac{49 π^{2}}{16}}$

Step 2.7

To write $\frac{9 π^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{9 π^{2}}{4} \cdot \frac{4}{4} + \frac{49 π^{2}}{16}}$

Step 2.8

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.8.1

Multiply $\frac{9 π^{2}}{4}$ by $\frac{4}{4}$ .

$\sqrt{\frac{9 π^{2} \cdot 4}{4 \cdot 4} + \frac{49 π^{2}}{16}}$

Step 2.8.2

Multiply $4$ by $4$ .

$\sqrt{\frac{9 π^{2} \cdot 4}{16} + \frac{49 π^{2}}{16}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{9 π^{2} \cdot 4 + 49 π^{2}}{16}}$

Step 2.10

Rewrite $\frac{9 π^{2} \cdot 4 + 49 π^{2}}{16}$ in a factored form.

Tap for more steps...

Step 2.10.1

Multiply $4$ by $9$ .

$\sqrt{\frac{36 π^{2} + 49 π^{2}}{16}}$

Step 2.10.2

Add $36 π^{2}$ and $49 π^{2}$ .

$\sqrt{\frac{85 π^{2}}{16}}$

Step 2.11

Rewrite $\sqrt{\frac{85 π^{2}}{16}}$ as $\frac{\sqrt{85 π^{2}}}{\sqrt{16}}$ .

$\frac{\sqrt{85 π^{2}}}{\sqrt{16}}$

Step 2.12

Simplify the numerator.

Tap for more steps...

Step 2.12.1

Reorder $85$ and $π^{2}$ .

$\frac{\sqrt{π^{2} \cdot 85}}{\sqrt{16}}$

Step 2.12.2

Pull terms out from under the radical.

$\frac{π \sqrt{85}}{\sqrt{16}}$

Step 2.13

Simplify the denominator.

Tap for more steps...

Step 2.13.1

Rewrite $16$ as $4^{2}$ .

$\frac{π \sqrt{85}}{\sqrt{4^{2}}}$

Step 2.13.2

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

$\frac{π \sqrt{85}}{4}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{3 π}{2}}{\frac{π \sqrt{85}}{4}}$ .

$\frac{\frac{3 π}{2}}{\frac{π \sqrt{85}}{4}}$

Step 4

Simplify $\cos (θ)$ .

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = \frac{3 π}{2} \cdot \frac{4}{π \sqrt{85}}$

Step 4.2

Cancel the common factor of $π$ .

Tap for more steps...

Step 4.2.1

Factor $π$ out of $3 π$ .

$\cos (θ) = \frac{π \cdot 3}{2} \cdot \frac{4}{π \sqrt{85}}$

Step 4.2.2

Factor $π$ out of $π \sqrt{85}$ .

$\cos (θ) = \frac{π \cdot 3}{2} \cdot \frac{4}{π (\sqrt{85})}$

Step 4.2.3

Cancel the common factor.

$\cos (θ) = \frac{π \cdot 3}{2} \cdot \frac{4}{π \sqrt{85}}$

Step 4.2.4

Rewrite the expression.

$\cos (θ) = \frac{3}{2} \cdot \frac{4}{\sqrt{85}}$

Step 4.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.1

Factor $2$ out of $4$ .

$\cos (θ) = \frac{3}{2} \cdot \frac{2 (2)}{\sqrt{85}}$

Step 4.3.2

Cancel the common factor.

$\cos (θ) = \frac{3}{2} \cdot \frac{2 \cdot 2}{\sqrt{85}}$

Step 4.3.3

Rewrite the expression.

$\cos (θ) = 3 (\frac{2}{\sqrt{85}})$

Step 4.4

Combine $3$ and $\frac{2}{\sqrt{85}}$ .

$\cos (θ) = \frac{3 \cdot 2}{\sqrt{85}}$

Step 4.5

Multiply $3$ by $2$ .

$\cos (θ) = \frac{6}{\sqrt{85}}$

Step 4.6

Multiply $\frac{6}{\sqrt{85}}$ by $\frac{\sqrt{85}}{\sqrt{85}}$ .

$\cos (θ) = \frac{6}{\sqrt{85}} \cdot \frac{\sqrt{85}}{\sqrt{85}}$

Step 4.7

Combine and simplify the denominator.

Tap for more steps...

Step 4.7.1

Multiply $\frac{6}{\sqrt{85}}$ by $\frac{\sqrt{85}}{\sqrt{85}}$ .

$\cos (θ) = \frac{6 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.7.2

Raise $\sqrt{85}$ to the power of $1$ .

$\cos (θ) = \frac{6 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.7.3

Raise $\sqrt{85}$ to the power of $1$ .

$\cos (θ) = \frac{6 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Step 4.7.4

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

$\cos (θ) = \frac{6 \sqrt{85}}{{\sqrt{85}}^{1 + 1}}$

Step 4.7.5

Add $1$ and $1$ .

$\cos (θ) = \frac{6 \sqrt{85}}{{\sqrt{85}}^{2}}$

Step 4.7.6

Rewrite ${\sqrt{85}}^{2}$ as $85$ .

Tap for more steps...

Step 4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{85}$ as $85^{\frac{1}{2}}$ .

$\cos (θ) = \frac{6 \sqrt{85}}{{(85^{\frac{1}{2}})}^{2}}$

Step 4.7.6.2

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

$\cos (θ) = \frac{6 \sqrt{85}}{85^{\frac{1}{2} \cdot 2}}$

Step 4.7.6.3

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

$\cos (θ) = \frac{6 \sqrt{85}}{85^{\frac{2}{2}}}$

Step 4.7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.7.6.4.1

Cancel the common factor.

$\cos (θ) = \frac{6 \sqrt{85}}{85^{\frac{2}{2}}}$

Step 4.7.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{6 \sqrt{85}}{85}$

Step 4.7.6.5

Evaluate the exponent.

$\cos (θ) = \frac{6 \sqrt{85}}{85}$

Step 5

Approximate the result.

$\cos (θ) = \frac{6 \sqrt{85}}{85} \approx 0.65079137$