Precalculus Examples

$\cos (x) = - \frac{60}{61}$

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.

$Opposite = - \sqrt{{(61)}^{2} - {(- 60)}^{2}}$

Step 4

Simplify inside the radical.

Tap for more steps...

Step 4.1

Negate $\sqrt{{(61)}^{2} - {(- 60)}^{2}}$ .

Opposite $= - \sqrt{{(61)}^{2} - {(- 60)}^{2}}$

Step 4.2

Raise $61$ to the power of $2$ .

Opposite $= - \sqrt{3721 - {(- 60)}^{2}}$

Step 4.3

Raise $- 60$ to the power of $2$ .

Opposite $= - \sqrt{3721 - 1 \cdot 3600}$

Step 4.4

Multiply $- 1$ by $3600$ .

Opposite $= - \sqrt{3721 - 3600}$

Step 4.5

Subtract $3600$ from $3721$ .

Opposite $= - \sqrt{121}$

Step 4.6

Rewrite $121$ as $11^{2}$ .

Opposite $= - \sqrt{11^{2}}$

Step 4.7

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

Opposite $= - 1 \cdot 11$

Step 4.8

Multiply $- 1$ by $11$ .

Opposite $= - 11$

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.

$\sin (x) = \frac{- 11}{61}$

Step 5.3

Move the negative in front of the fraction.

$\sin (x) = - \frac{11}{61}$

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.

$\tan (x) = \frac{- 11}{- 60}$

Step 6.3

Dividing two negative values results in a positive value.

$\tan (x) = \frac{11}{60}$

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.

$\cot (x) = \frac{- 60}{- 11}$

Step 7.3

Dividing two negative values results in a positive value.

$\cot (x) = \frac{60}{11}$

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.

$\sec (x) = \frac{61}{- 60}$

Step 8.3

Move the negative in front of the fraction.

$\sec (x) = - \frac{61}{60}$

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.

$\csc (x) = \frac{61}{- 11}$

Step 9.3

Move the negative in front of the fraction.

$\csc (x) = - \frac{61}{11}$

Step 10

This is the solution to each trig value.

$\sin (x) = - \frac{11}{61}$

$\cos (x) = - \frac{60}{61}$

$\tan (x) = \frac{11}{60}$

$\cot (x) = \frac{60}{11}$

$\sec (x) = - \frac{61}{60}$

$\csc (x) = - \frac{61}{11}$