Trigonometry Examples

(- \sqrt{3}, 1)

$(- \sqrt{3}, 1)$

Step 1

To find the

csc (θ)

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

(0, 0)

$(0, 0)$ and

(- \sqrt{3}, 1)

$(- \sqrt{3}, 1)$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(- \sqrt{3}, 0)

$(- \sqrt{3}, 0)$ , and

(- \sqrt{3}, 1)

$(- \sqrt{3}, 1)$ .

Opposite :

1

$1$

Adjacent :

- \sqrt{3}

$- \sqrt{3}$

Step 2

Find the hypotenuse using Pythagorean theorem

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

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

Tap for more steps...

Step 2.1

Simplify the expression.

Tap for more steps...

Step 2.1.1

Apply the product rule to

- \sqrt{3}

$- \sqrt{3}$ .

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

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

Step 2.1.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 2.1.3

Multiply

{\sqrt{3}}^{2}

${\sqrt{3}}^{2}$ by

1

$1$ .

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

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

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

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

Step 2.2

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

Tap for more steps...

Step 2.2.1

Use

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

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

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

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

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

Step 2.2.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 2.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 2.2.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

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

Step 2.2.5

Evaluate the exponent.

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

Step 2.3

Simplify the expression.

Tap for more steps...

Step 2.3.1

One to any power is one.

$\sqrt{3 + 1}$

Step 2.3.2

Add

$3$ and

$1$ .

$\sqrt{4}$

Step 2.3.3

Rewrite

$4$ as

$2^{2}$ .

$\sqrt{2^{2}}$

Step 2.4

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

$2$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore

$\csc (θ) = \frac{2}{1}$ .

$\frac{2}{1}$

Step 4

Divide

$2$ by

$1$ .

$\csc (θ) = 2$