Trigonometry Examples

$(7, 1)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(7, 1)$ , draw the triangle between the three points $(0, 0)$ , $(7, 0)$ , and $(7, 1)$ .

Opposite : $1$

Adjacent : $7$

Step 2

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

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Step 2.1

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

$\sqrt{49 + {(1)}^{2}}$

Step 2.2

One to any power is one.

$\sqrt{49 + 1}$

Step 2.3

Add $49$ and $1$ .

$\sqrt{50}$

Step 2.4

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 2.4.1

Factor $25$ out of $50$ .

$\sqrt{25 (2)}$

Step 2.4.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} \cdot 2}$

Step 2.5

Pull terms out from under the radical.

$5 \sqrt{2}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{7}{5 \sqrt{2}}$ .

$\frac{7}{5 \sqrt{2}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply $\frac{7}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (θ) = \frac{7}{5 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{7}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (θ) = \frac{7 \sqrt{2}}{5 \sqrt{2} \sqrt{2}}$

Step 4.2.2

Move $\sqrt{2}$ .

$\cos (θ) = \frac{7 \sqrt{2}}{5 (\sqrt{2} \sqrt{2})}$

Step 4.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (θ) = \frac{7 \sqrt{2}}{5 (\sqrt{2} \sqrt{2})}$

Step 4.2.4

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (θ) = \frac{7 \sqrt{2}}{5 (\sqrt{2} \sqrt{2})}$

Step 4.2.5

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

$\cos (θ) = \frac{7 \sqrt{2}}{5 {\sqrt{2}}^{1 + 1}}$

Step 4.2.6

Add $1$ and $1$ .

$\cos (θ) = \frac{7 \sqrt{2}}{5 {\sqrt{2}}^{2}}$

Step 4.2.7

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

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Step 4.2.7.1

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

$\cos (θ) = \frac{7 \sqrt{2}}{5 {(2^{\frac{1}{2}})}^{2}}$

Step 4.2.7.2

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

$\cos (θ) = \frac{7 \sqrt{2}}{5 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 4.2.7.3

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

$\cos (θ) = \frac{7 \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 4.2.7.4

Cancel the common factor of $2$ .

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Step 4.2.7.4.1

Cancel the common factor.

$\cos (θ) = \frac{7 \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 4.2.7.4.2

Rewrite the expression.

$\cos (θ) = \frac{7 \sqrt{2}}{5 \cdot 2}$

Step 4.2.7.5

Evaluate the exponent.

$\cos (θ) = \frac{7 \sqrt{2}}{5 \cdot 2}$

Step 4.3

Multiply $5$ by $2$ .

$\cos (θ) = \frac{7 \sqrt{2}}{10}$

Step 5

Approximate the result.

$\cos (θ) = \frac{7 \sqrt{2}}{10} \approx 0.98994949$