Trigonometry Examples

$(0.791, 1)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(0.791, 1)$ , draw the triangle between the three points $(0, 0)$ , $(0.791, 0)$ , and $(0.791, 1)$ .

Opposite : $1$

Adjacent : $0.791$

Step 2

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

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Step 2.1

Raise $0.791$ to the power of $2$ .

$\sqrt{0.625681 + {(1)}^{2}}$

Step 2.2

One to any power is one.

$\sqrt{0.625681 + 1}$

Step 2.3

Add $0.625681$ and $1$ .

$\sqrt{1.625681}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{1}{\sqrt{1.625681}}$ .

$\frac{1}{\sqrt{1.625681}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply $\frac{1}{\sqrt{1.625681}}$ by $\frac{\sqrt{1.625681}}{\sqrt{1.625681}}$ .

$\sin (θ) = \frac{1}{\sqrt{1.625681}} \cdot \frac{\sqrt{1.625681}}{\sqrt{1.625681}}$

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

Multiply $\frac{1}{\sqrt{1.625681}}$ by $\frac{\sqrt{1.625681}}{\sqrt{1.625681}}$ .

$\sin (θ) = \frac{\sqrt{1.625681}}{\sqrt{1.625681} \sqrt{1.625681}}$

Step 4.2.2

Raise $\sqrt{1.625681}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{1.625681}}{\sqrt{1.625681} \sqrt{1.625681}}$

Step 4.2.3

Raise $\sqrt{1.625681}$ to the power of $1$ .

$\sin (θ) = \frac{\sqrt{1.625681}}{\sqrt{1.625681} \sqrt{1.625681}}$

Step 4.2.4

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

$\sin (θ) = \frac{\sqrt{1.625681}}{{\sqrt{1.625681}}^{1 + 1}}$

Step 4.2.5

Add $1$ and $1$ .

$\sin (θ) = \frac{\sqrt{1.625681}}{{\sqrt{1.625681}}^{2}}$

Step 4.2.6

Rewrite ${\sqrt{1.625681}}^{2}$ as $1.625681$ .

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Step 4.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1.625681}$ as ${1.625681}^{\frac{1}{2}}$ .

$\sin (θ) = \frac{\sqrt{1.625681}}{{({1.625681}^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

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

$\sin (θ) = \frac{\sqrt{1.625681}}{{1.625681}^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

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

$\sin (θ) = \frac{\sqrt{1.625681}}{{1.625681}^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.6.4.1

Cancel the common factor.

$\sin (θ) = \frac{\sqrt{1.625681}}{{1.625681}^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\sin (θ) = \frac{\sqrt{1.625681}}{1.625681}$

Step 4.2.6.5

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{1.625681}}{1.625681}$

Step 4.3

Evaluate the root.

$\sin (θ) = \frac{1.27502196}{1.625681}$

Step 4.4

Divide $1.27502196$ by $1.625681$ .

$\sin (θ) = 0.78430021$