Trigonometry Examples

(1, 3)

$(1, 3)$

Step 1

To find the

cos (θ)

$\cos (θ)$ between the x-axis and the line between the points

(0, 0)

$(0, 0)$ and

(1, 3)

$(1, 3)$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(1, 0)

$(1, 0)$ , and

(1, 3)

$(1, 3)$ .

Opposite :

3

$3$

Adjacent :

1

$1$

Step 2

Find the hypotenuse using Pythagorean theorem

c = \sqrt{a^{2} + b^{2}}

$c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

One to any power is one.

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

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

Step 2.2

Raise

3

$3$ to the power of

2

$2$ .

\sqrt{1 + 9}

$\sqrt{1 + 9}$

Step 2.3

Add

1

$1$ and

9

$9$ .

\sqrt{10}

$\sqrt{10}$

\sqrt{10}

$\sqrt{10}$

Step 3

cos (θ) = \frac{Adjacent}{Hypotenuse}

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore

cos (θ) = \frac{1}{\sqrt{10}}

$\cos (θ) = \frac{1}{\sqrt{10}}$ .

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

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

Step 4

Simplify

cos (θ)

$\cos (θ)$ .

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Step 4.1

Multiply

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

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

\frac{\sqrt{10}}{\sqrt{10}}

$\frac{\sqrt{10}}{\sqrt{10}}$ .

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

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

\frac{\sqrt{10}}{\sqrt{10}}

$\frac{\sqrt{10}}{\sqrt{10}}$ .

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

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

Step 4.2.2

Raise

\sqrt{10}

$\sqrt{10}$ to the power of

1

$1$ .

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

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

Step 4.2.3

Raise

\sqrt{10}

$\sqrt{10}$ to the power of

1

$1$ .

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

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

Step 4.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 4.2.5

Add

1

$1$ and

1

$1$ .

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

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

Step 4.2.6

Rewrite

{\sqrt{10}}^{2}

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

10

$10$ .

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Step 4.2.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{10}

$\sqrt{10}$ as

10^{\frac{1}{2}}

$10^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 4.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 4.2.6.4

Cancel the common factor of

2

$2$ .

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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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