Trigonometry Examples

$(- 2, \sqrt{3})$

Step 1

To find the $\sec (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 2, \sqrt{3})$ , draw the triangle between the three points $(0, 0)$ , $(- 2, 0)$ , and $(- 2, \sqrt{3})$ .

Opposite : $\sqrt{3}$

Adjacent : $- 2$

Step 2

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

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Step 2.1

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

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

Step 2.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{4 + {(3^{\frac{1}{2}})}^{2}}$

Step 2.2.2

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

$\sqrt{4 + 3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

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

$\sqrt{4 + 3^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$\sqrt{4 + 3^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$\sqrt{4 + 3^{1}}$

Step 2.2.5

Evaluate the exponent.

$\sqrt{4 + 3}$

Step 2.3

Add $4$ and $3$ .

$\sqrt{7}$

Step 3

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

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

Step 4

Move the negative in front of the fraction.

$\sec (θ) = - \frac{\sqrt{7}}{2}$

Step 5

Approximate the result.

$\sec (θ) = - \frac{\sqrt{7}}{2} \approx - 1.32287565$