Trigonometry Examples

$\cos (x) = \frac{9}{41}$

Step 1

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

$\cos (x) = \frac{adjacent}{hypotenuse}$

Step 2

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.

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

Step 3

Replace the known values in the equation.

$Opposite = \sqrt{{(41)}^{2} - {(9)}^{2}}$

Step 4

Simplify inside the radical.

Tap for more steps...

Step 4.1

Raise $41$ to the power of $2$ .

Opposite $= \sqrt{1681 - {(9)}^{2}}$

Step 4.2

Raise $9$ to the power of $2$ .

Opposite $= \sqrt{1681 - 1 \cdot 81}$

Step 4.3

Multiply $- 1$ by $81$ .

Opposite $= \sqrt{1681 - 81}$

Step 4.4

Subtract $81$ from $1681$ .

Opposite $= \sqrt{1600}$

Step 4.5

Rewrite $1600$ as $40^{2}$ .

Opposite $= \sqrt{40^{2}}$

Step 4.6

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

Opposite $= 40$

Step 5

Find the value of sine.

Tap for more steps...

Step 5.1

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

$\sin (x) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (x) = \frac{40}{41}$

Step 6

Find the value of tangent.

Tap for more steps...

Step 6.1

Use the definition of tangent to find the value of $\tan (x)$ .

$\tan (x) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

$\tan (x) = \frac{40}{9}$

Step 7

Find the value of cotangent.

Tap for more steps...

Step 7.1

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

$\cot (x) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (x) = \frac{9}{40}$

Step 8

Find the value of secant.

Tap for more steps...

Step 8.1

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

$\sec (x) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (x) = \frac{41}{9}$

Step 9

Find the value of cosecant.

Tap for more steps...

Step 9.1

Use the definition of cosecant to find the value of $\csc (x)$ .

$\csc (x) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (x) = \frac{41}{40}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{40}{41}$

$\cos (x) = \frac{9}{41}$

$\tan (x) = \frac{40}{9}$

$\cot (x) = \frac{9}{40}$

$\sec (x) = \frac{41}{9}$

$\csc (x) = \frac{41}{40}$