Trigonometry Examples

sec (θ) = - 3

$\sec (θ) = - 3$ ,

tan (θ) > 0

$\tan (θ) > 0$

Step 1

The tangent function is positive in the first and third quadrants. The secant function is negative in the second and third quadrants. The set of solutions for

θ

$θ$ are limited to the third quadrant since that is the only quadrant found in both sets.

Solution is in the third quadrant.

Step 2

Use the definition of secant to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

sec (θ) = \frac{hypotenuse}{adjacent}

$\sec (θ) = \frac{hypotenuse}{adjacent}$

Step 3

Find the opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.

Opposite = - \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}

$Opposite = - \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Step 4

Replace the known values in the equation.

Opposite = - \sqrt{{(3)}^{2} - {(- 1)}^{2}}

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

Step 5

Simplify inside the radical.

Tap for more steps...

Step 5.1

Negate

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

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

Opposite

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

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

Step 5.2

Raise

3

$3$ to the power of

2

$2$ .

Opposite

= - \sqrt{9 - {(- 1)}^{2}}

$= - \sqrt{9 - {(- 1)}^{2}}$

Step 5.3

Multiply

- 1

$- 1$ by

{(- 1)}^{2}

${(- 1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 5.3.1

Multiply

- 1

$- 1$ by

{(- 1)}^{2}

${(- 1)}^{2}$ .

Tap for more steps...

Step 5.3.1.1

Raise

- 1

$- 1$ to the power of

1

$1$ .

Opposite

= - \sqrt{9 + (- 1) {(- 1)}^{2}}

$= - \sqrt{9 + (- 1) {(- 1)}^{2}}$

Step 5.3.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

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

Opposite

= - \sqrt{9 + {(- 1)}^{1 + 2}}

$= - \sqrt{9 + {(- 1)}^{1 + 2}}$

Opposite

= - \sqrt{9 + {(- 1)}^{1 + 2}}

$= - \sqrt{9 + {(- 1)}^{1 + 2}}$

Step 5.3.2

Add

1

$1$ and

2

$2$ .

Opposite

= - \sqrt{9 + {(- 1)}^{3}}

$= - \sqrt{9 + {(- 1)}^{3}}$

Opposite

= - \sqrt{9 + {(- 1)}^{3}}

$= - \sqrt{9 + {(- 1)}^{3}}$

Step 5.4

Raise

- 1

$- 1$ to the power of

3

$3$ .

Opposite

= - \sqrt{9 - 1}

$= - \sqrt{9 - 1}$

Step 5.5

Subtract

1

$1$ from

9

$9$ .

Opposite

= - \sqrt{8}

$= - \sqrt{8}$

Step 5.6

Rewrite

8

$8$ as

2^{2} \cdot 2

$2^{2} \cdot 2$ .

Tap for more steps...

Step 5.6.1

Factor

4

$4$ out of

8

$8$ .

Opposite

= - \sqrt{4 (2)}

$= - \sqrt{4 (2)}$

Step 5.6.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

Opposite

= - \sqrt{2^{2} \cdot 2}

$= - \sqrt{2^{2} \cdot 2}$

Opposite

= - \sqrt{2^{2} \cdot 2}

$= - \sqrt{2^{2} \cdot 2}$

Step 5.7

Pull terms out from under the radical.

Opposite

= - (2 \sqrt{2})

$= - (2 \sqrt{2})$

Step 5.8

Multiply

2

$2$ by

- 1

$- 1$ .

Opposite

= - 2 \sqrt{2}

$= - 2 \sqrt{2}$

Opposite

= - 2 \sqrt{2}

$= - 2 \sqrt{2}$

Step 6

Find the value of sine.

Tap for more steps...

Step 6.1

Use the definition of sine to find the value of

sin (θ)

$\sin (θ)$ .

sin (θ) = \frac{opp}{hyp}

$\sin (θ) = \frac{opp}{hyp}$

Step 6.2

Substitute in the known values.

sin (θ) = \frac{- 2 \sqrt{2}}{3}

$\sin (θ) = \frac{- 2 \sqrt{2}}{3}$

Step 6.3

Move the negative in front of the fraction.

sin (θ) = - \frac{2 \sqrt{2}}{3}

$\sin (θ) = - \frac{2 \sqrt{2}}{3}$

sin (θ) = - \frac{2 \sqrt{2}}{3}

$\sin (θ) = - \frac{2 \sqrt{2}}{3}$

Step 7

Find the value of cosine.

Tap for more steps...

Step 7.1

Use the definition of cosine to find the value of

cos (θ)

$\cos (θ)$ .

cos (θ) = \frac{adj}{hyp}

$\cos (θ) = \frac{adj}{hyp}$

Step 7.2

Substitute in the known values.

cos (θ) = \frac{- 1}{3}

$\cos (θ) = \frac{- 1}{3}$

Step 7.3

Move the negative in front of the fraction.

cos (θ) = - \frac{1}{3}

$\cos (θ) = - \frac{1}{3}$

cos (θ) = - \frac{1}{3}

$\cos (θ) = - \frac{1}{3}$

Step 8

Find the value of tangent.

Tap for more steps...

Step 8.1

Use the definition of tangent to find the value of

tan (θ)

$\tan (θ)$ .

tan (θ) = \frac{opp}{adj}

$\tan (θ) = \frac{opp}{adj}$

Step 8.2

Substitute in the known values.

tan (θ) = \frac{- 2 \sqrt{2}}{- 1}

$\tan (θ) = \frac{- 2 \sqrt{2}}{- 1}$

Step 8.3

Simplify the value of

tan (θ)

$\tan (θ)$ .

Tap for more steps...

Step 8.3.1

Move the negative one from the denominator of

\frac{- 2 \sqrt{2}}{- 1}

$\frac{- 2 \sqrt{2}}{- 1}$ .

tan (θ) = - 1 \cdot (- 2 \sqrt{2})

$\tan (θ) = - 1 \cdot (- 2 \sqrt{2})$

Step 8.3.2

Rewrite

- 1 \cdot (- 2 \sqrt{2})

$- 1 \cdot (- 2 \sqrt{2})$ as

- (- 2 \sqrt{2})

$- (- 2 \sqrt{2})$ .

tan (θ) = - (- 2 \sqrt{2})

$\tan (θ) = - (- 2 \sqrt{2})$

Step 8.3.3

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

tan (θ) = 2 \sqrt{2}

$\tan (θ) = 2 \sqrt{2}$

tan (θ) = 2 \sqrt{2}

$\tan (θ) = 2 \sqrt{2}$

tan (θ) = 2 \sqrt{2}

$\tan (θ) = 2 \sqrt{2}$

Step 9

Find the value of cotangent.

Tap for more steps...

Step 9.1

Use the definition of cotangent to find the value of

cot (θ)

$\cot (θ)$ .

cot (θ) = \frac{adj}{opp}

$\cot (θ) = \frac{adj}{opp}$

Step 9.2

Substitute in the known values.

cot (θ) = \frac{- 1}{- 2 \sqrt{2}}

$\cot (θ) = \frac{- 1}{- 2 \sqrt{2}}$

Step 9.3

Simplify the value of

cot (θ)

$\cot (θ)$ .

Tap for more steps...

Step 9.3.1

Dividing two negative values results in a positive value.

cot (θ) = \frac{1}{2 \sqrt{2}}

$\cot (θ) = \frac{1}{2 \sqrt{2}}$

Step 9.3.2

Multiply

\frac{1}{2 \sqrt{2}}

$\frac{1}{2 \sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

cot (θ) = \frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}

$\cot (θ) = \frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 9.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 9.3.3.1

Multiply

\frac{1}{2 \sqrt{2}}

$\frac{1}{2 \sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

cot (θ) = \frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}

$\cot (θ) = \frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 9.3.3.2

Move

\sqrt{2}

$\sqrt{2}$ .

cot (θ) = \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}

$\cot (θ) = \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 9.3.3.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

cot (θ) = \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}

$\cot (θ) = \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 9.3.3.4

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

cot (θ) = \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}

$\cot (θ) = \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 9.3.3.5

Use the power rule

a^{m} a^{n} = a^{m + n}

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

cot (θ) = \frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}

$\cot (θ) = \frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 9.3.3.6

Add

1

$1$ and

1

$1$ .

cot (θ) = \frac{\sqrt{2}}{2 {\sqrt{2}}^{2}}

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

Step 9.3.3.7

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

Tap for more steps...

Step 9.3.3.7.1

Use

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

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

\sqrt{2}

$\sqrt{2}$ as

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

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

cot (θ) = \frac{\sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}

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

Step 9.3.3.7.2

Apply the power rule and multiply exponents,

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

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

cot (θ) = \frac{\sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}

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

Step 9.3.3.7.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

cot (θ) = \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}

$\cot (θ) = \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 9.3.3.7.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 9.3.3.7.4.1

Cancel the common factor.

$\cot (θ) = \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 9.3.3.7.4.2

Rewrite the expression.

$\cot (θ) = \frac{\sqrt{2}}{2 \cdot 2}$

Step 9.3.3.7.5

Evaluate the exponent.

$\cot (θ) = \frac{\sqrt{2}}{2 \cdot 2}$

Step 9.3.4

Multiply

$2$ by

$2$ .

$\cot (θ) = \frac{\sqrt{2}}{4}$

Step 10

Find the value of cosecant.

Tap for more steps...

Step 10.1

Use the definition of cosecant to find the value of

$\csc (θ)$ .

$\csc (θ) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

$\csc (θ) = \frac{3}{- 2 \sqrt{2}}$

Step 10.3

Simplify the value of

$\csc (θ)$ .

Tap for more steps...

Step 10.3.1

Move the negative in front of the fraction.

$\csc (θ) = - \frac{3}{2 \sqrt{2}}$

Step 10.3.2

Multiply

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

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\csc (θ) = - (\frac{3}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 10.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 10.3.3.1

Multiply

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

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 10.3.3.2

Move

$\sqrt{2}$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 10.3.3.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 10.3.3.4

Raise

$\sqrt{2}$ to the power of

$1$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 10.3.3.5

Use the power rule

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

$\csc (θ) = - \frac{3 \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 10.3.3.6

Add

$1$ and

$1$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 10.3.3.7

Rewrite

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

$2$ .

Tap for more steps...

Step 10.3.3.7.1

Use

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

$\sqrt{2}$ as

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

$\csc (θ) = - \frac{3 \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 10.3.3.7.2

Apply the power rule and multiply exponents,

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

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 10.3.3.7.3

Combine

$\frac{1}{2}$ and

$2$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 10.3.3.7.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 10.3.3.7.4.1

Cancel the common factor.

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 10.3.3.7.4.2

Rewrite the expression.

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2}$

Step 10.3.3.7.5

Evaluate the exponent.

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2}$

Step 10.3.4

Multiply

$2$ by

$2$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{4}$

Step 11

This is the solution to each trig value.

$\sin (θ) = - \frac{2 \sqrt{2}}{3}$

$\cos (θ) = - \frac{1}{3}$

$\tan (θ) = 2 \sqrt{2}$

$\cot (θ) = \frac{\sqrt{2}}{4}$

$\sec (θ) = - 3$

$\csc (θ) = - \frac{3 \sqrt{2}}{4}$