Trigonometry Examples

$\tan (θ) = \frac{3}{4}$ , $\sin (θ) < 0$

Step 1

The sine function is negative in the third and fourth quadrants. The tangent function is positive in the first and third quadrants. The set of solutions for $θ$ 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 tangent to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\tan (θ) = \frac{opposite}{adjacent}$

Step 3

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

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Step 4

Replace the known values in the equation.

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

Step 5

Simplify inside the radical.

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Step 5.1

Raise $- 3$ to the power of $2$ .

Hypotenuse $= \sqrt{9 + {(- 4)}^{2}}$

Step 5.2

Raise $- 4$ to the power of $2$ .

Hypotenuse $= \sqrt{9 + 16}$

Step 5.3

Add $9$ and $16$ .

Hypotenuse $= \sqrt{25}$

Step 5.4

Rewrite $25$ as $5^{2}$ .

Hypotenuse $= \sqrt{5^{2}}$

Step 5.5

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

Hypotenuse $= 5$

Step 6

Find the value of sine.

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Step 6.1

Use the definition of sine to find the value of $\sin (θ)$ .

$\sin (θ) = \frac{opp}{hyp}$

Step 6.2

Substitute in the known values.

$\sin (θ) = \frac{- 3}{5}$

Step 6.3

Move the negative in front of the fraction.

$\sin (θ) = - \frac{3}{5}$

Step 7

Find the value of cosine.

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Step 7.1

Use the definition of cosine to find the value of $\cos (θ)$ .

$\cos (θ) = \frac{adj}{hyp}$

Step 7.2

Substitute in the known values.

$\cos (θ) = \frac{- 4}{5}$

Step 7.3

Move the negative in front of the fraction.

$\cos (θ) = - \frac{4}{5}$

Step 8

Find the value of cotangent.

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Step 8.1

Use the definition of cotangent to find the value of $\cot (θ)$ .

$\cot (θ) = \frac{adj}{opp}$

Step 8.2

Substitute in the known values.

$\cot (θ) = \frac{- 4}{- 3}$

Step 8.3

Dividing two negative values results in a positive value.

$\cot (θ) = \frac{4}{3}$

Step 9

Find the value of secant.

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Step 9.1

Use the definition of secant to find the value of $\sec (θ)$ .

$\sec (θ) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

$\sec (θ) = \frac{5}{- 4}$

Step 9.3

Move the negative in front of the fraction.

$\sec (θ) = - \frac{5}{4}$

Step 10

Find the value of cosecant.

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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.

$\csc (θ) = \frac{5}{- 3}$

Step 10.3

Move the negative in front of the fraction.

$\csc (θ) = - \frac{5}{3}$

Step 11

This is the solution to each trig value.

$\sin (θ) = - \frac{3}{5}$

$\cos (θ) = - \frac{4}{5}$

$\tan (θ) = \frac{3}{4}$

$\cot (θ) = \frac{4}{3}$

$\sec (θ) = - \frac{5}{4}$

$\csc (θ) = - \frac{5}{3}$