Trigonometry Examples

$(\sqrt{7}, 3)$

Step 1

To find the

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

$(0, 0)$ and

$(\sqrt{7}, 3)$ , draw the triangle between the three points

$(0, 0)$ ,

$(\sqrt{7}, 0)$ , and

$(\sqrt{7}, 3)$ .

Opposite :

$3$

Adjacent :

$\sqrt{7}$

Step 2

Find the hypotenuse using Pythagorean theorem

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

Tap for more steps...

Step 2.1

Rewrite

${\sqrt{7}}^{2}$ as

$7$ .

Tap for more steps...

Step 2.1.1

Use

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

$\sqrt{7}$ as

$7^{\frac{1}{2}}$ .

$\sqrt{{(7^{\frac{1}{2}})}^{2} + {(3)}^{2}}$

Step 2.1.2

Apply the power rule and multiply exponents,

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

$\sqrt{7^{\frac{1}{2} \cdot 2} + {(3)}^{2}}$

Step 2.1.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{7^{\frac{2}{2}} + {(3)}^{2}}$

Step 2.1.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 2.1.4.1

Cancel the common factor.

$\sqrt{7^{\frac{2}{2}} + {(3)}^{2}}$

Step 2.1.4.2

Rewrite the expression.

$\sqrt{7^{1} + {(3)}^{2}}$

Step 2.1.5

Evaluate the exponent.

$\sqrt{7 + {(3)}^{2}}$

Step 2.2

Simplify the expression.

Tap for more steps...

Step 2.2.1

Raise

$3$ to the power of

$2$ .

$\sqrt{7 + 9}$

Step 2.2.2

Add

$7$ and

$9$ .

$\sqrt{16}$

Step 2.2.3

Rewrite

$16$ as

$4^{2}$ .

$\sqrt{4^{2}}$

Step 2.2.4

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

$4$

Step 3

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

$\sec (θ) = \frac{4}{\sqrt{7}}$ .

$\frac{4}{\sqrt{7}}$

Step 4

Simplify

$\sec (θ)$ .

Tap for more steps...

Step 4.1

Multiply

$\frac{4}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sec (θ) = \frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4.2

Combine and simplify the denominator.

Tap for more steps...

Step 4.2.1

Multiply

$\frac{4}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sec (θ) = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.2.2

Raise

$\sqrt{7}$ to the power of

$1$ .

$\sec (θ) = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.2.3

Raise

$\sqrt{7}$ to the power of

$1$ .

$\sec (θ) = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 4.2.4

Use the power rule

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

$\sec (θ) = \frac{4 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 4.2.5

Add

$1$ and

$1$ .

$\sec (θ) = \frac{4 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 4.2.6

Rewrite

${\sqrt{7}}^{2}$ as

$7$ .

Tap for more steps...

Step 4.2.6.1

Use

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

$\sqrt{7}$ as

$7^{\frac{1}{2}}$ .

$\sec (θ) = \frac{4 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 4.2.6.2

Apply the power rule and multiply exponents,

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

$\sec (θ) = \frac{4 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 4.2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sec (θ) = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.2.6.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 4.2.6.4.1

Cancel the common factor.

$\sec (θ) = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 4.2.6.4.2

Rewrite the expression.

$\sec (θ) = \frac{4 \sqrt{7}}{7}$

Step 4.2.6.5

Evaluate the exponent.

$\sec (θ) = \frac{4 \sqrt{7}}{7}$

Step 5

Approximate the result.

$\sec (θ) = \frac{4 \sqrt{7}}{7} \approx 1.51185789$