Trigonometry Examples

$(1, \frac{π}{3})$

Step 1

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

Opposite : $\frac{π}{3}$

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.

$\sqrt{1 + {(\frac{π}{3})}^{2}}$

Step 2.2

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

$\sqrt{1 + \frac{π^{2}}{3^{2}}}$

Step 2.3

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

$\sqrt{1 + \frac{π^{2}}{9}}$

Step 2.4

Write $1$ as a fraction with a common denominator.

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

Step 2.5

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + π^{2}}{9}}$

Step 2.6

Rewrite $\sqrt{\frac{9 + π^{2}}{9}}$ as $\frac{\sqrt{9 + π^{2}}}{\sqrt{9}}$ .

$\frac{\sqrt{9 + π^{2}}}{\sqrt{9}}$

Step 2.7

Simplify the denominator.

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Step 2.7.1

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{9 + π^{2}}}{\sqrt{3^{2}}}$

Step 2.7.2

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

$\frac{\sqrt{9 + π^{2}}}{3}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{1}{\frac{\sqrt{9 + π^{2}}}{3}}$ .

$\frac{1}{\frac{\sqrt{9 + π^{2}}}{3}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = 1 (\frac{3}{\sqrt{9 + π^{2}}})$

Step 4.2

Multiply $\frac{3}{\sqrt{9 + π^{2}}}$ by $1$ .

$\cos (θ) = \frac{3}{\sqrt{9 + π^{2}}}$

Step 4.3

Multiply $\frac{3}{\sqrt{9 + π^{2}}}$ by $\frac{\sqrt{9 + π^{2}}}{\sqrt{9 + π^{2}}}$ .

$\cos (θ) = \frac{3}{\sqrt{9 + π^{2}}} \cdot \frac{\sqrt{9 + π^{2}}}{\sqrt{9 + π^{2}}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{3}{\sqrt{9 + π^{2}}}$ by $\frac{\sqrt{9 + π^{2}}}{\sqrt{9 + π^{2}}}$ .

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{\sqrt{9 + π^{2}} \sqrt{9 + π^{2}}}$

Step 4.4.2

Raise $\sqrt{9 + π^{2}}$ to the power of $1$ .

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{\sqrt{9 + π^{2}} \sqrt{9 + π^{2}}}$

Step 4.4.3

Raise $\sqrt{9 + π^{2}}$ to the power of $1$ .

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{\sqrt{9 + π^{2}} \sqrt{9 + π^{2}}}$

Step 4.4.4

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

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{{\sqrt{9 + π^{2}}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{{\sqrt{9 + π^{2}}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{9 + π^{2}}}^{2}$ as $9 + π^{2}$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + π^{2}}$ as ${(9 + π^{2})}^{\frac{1}{2}}$ .

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{{({(9 + π^{2})}^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{{(9 + π^{2})}^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{{(9 + π^{2})}^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{{(9 + π^{2})}^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{9 + π^{2}}$

Step 4.4.6.5

Simplify.

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{9 + π^{2}}$

Step 5

Approximate the result.

$\cos (θ) = \frac{3 \sqrt{9 + π^{2}}}{9 + π^{2}} \approx 0.69062112$