Trigonometry Examples

cot (θ) = \frac{12}{5}

$\cot (θ) = \frac{12}{5}$ ,

sin (θ) > 0

$\sin (θ) > 0$

Step 1

The sine function is positive in the first and second quadrants. The cotangent function is positive in the first and third quadrants. The set of solutions for

θ

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

Solution is in the first quadrant.

Step 2

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

cot (θ) = \frac{adjacent}{opposite}

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

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}}

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

Step 4

Replace the known values in the equation.

Hypotenuse = \sqrt{{(5)}^{2} + {(12)}^{2}}

$Hypotenuse = \sqrt{{(5)}^{2} + {(12)}^{2}}$

Step 5

Simplify inside the radical.

Tap for more steps...

Step 5.1

Raise

5

$5$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{25 + {(12)}^{2}}

$= \sqrt{25 + {(12)}^{2}}$

Step 5.2

Raise

12

$12$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{25 + 144}

$= \sqrt{25 + 144}$

Step 5.3

Add

25

$25$ and

144

$144$ .

Hypotenuse

= \sqrt{169}

$= \sqrt{169}$

Step 5.4

Rewrite

169

$169$ as

13^{2}

$13^{2}$ .

Hypotenuse

= \sqrt{13^{2}}

$= \sqrt{13^{2}}$

Step 5.5

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

Hypotenuse

= 13

$= 13$

Hypotenuse

= 13

$= 13$

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{5}{13}

$\sin (θ) = \frac{5}{13}$

sin (θ) = \frac{5}{13}

$\sin (θ) = \frac{5}{13}$

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{12}{13}

$\cos (θ) = \frac{12}{13}$

cos (θ) = \frac{12}{13}

$\cos (θ) = \frac{12}{13}$

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{5}{12}

$\tan (θ) = \frac{5}{12}$

tan (θ) = \frac{5}{12}

$\tan (θ) = \frac{5}{12}$

Step 9

Find the value of secant.

Tap for more steps...

Step 9.1

Use the definition of secant to find the value of

sec (θ)

$\sec (θ)$ .

sec (θ) = \frac{hyp}{adj}

$\sec (θ) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

sec (θ) = \frac{13}{12}

$\sec (θ) = \frac{13}{12}$

sec (θ) = \frac{13}{12}

$\sec (θ) = \frac{13}{12}$

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)

$\csc (θ)$ .

csc (θ) = \frac{hyp}{opp}

$\csc (θ) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

csc (θ) = \frac{13}{5}

$\csc (θ) = \frac{13}{5}$

csc (θ) = \frac{13}{5}

$\csc (θ) = \frac{13}{5}$

Step 11

This is the solution to each trig value.

sin (θ) = \frac{5}{13}

$\sin (θ) = \frac{5}{13}$

cos (θ) = \frac{12}{13}

$\cos (θ) = \frac{12}{13}$

tan (θ) = \frac{5}{12}

$\tan (θ) = \frac{5}{12}$

cot (θ) = \frac{12}{5}

$\cot (θ) = \frac{12}{5}$

sec (θ) = \frac{13}{12}

$\sec (θ) = \frac{13}{12}$

csc (θ) = \frac{13}{5}

$\csc (θ) = \frac{13}{5}$