Find the Other Trig Values in Quadrant II cos(x)=-8/9

$\cos (x) = - \frac{8}{9}$

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.

$Opposite = \sqrt{{(9)}^{2} - {(- 8)}^{2}}$

Step 4

Simplify inside the radical.

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

Opposite $= \sqrt{81 - {(- 8)}^{2}}$

Raise $- 8$ to the power of $2$ .

Opposite $= \sqrt{81 - 1 \cdot 64}$

Multiply $- 1$ by $64$ .

Opposite $= \sqrt{81 - 64}$

Subtract $64$ from $81$ .

Opposite $= \sqrt{17}$

Step 5

Find the value of sine.

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

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

Substitute in the known values.

$\sin (x) = \frac{\sqrt{17}}{9}$

Step 6

Find the value of tangent.

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

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

Substitute in the known values.

$\tan (x) = \frac{\sqrt{17}}{- 8}$

Move the negative in front of the fraction.

$\tan (x) = - \frac{\sqrt{17}}{8}$

Step 7

Find the value of cotangent.

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

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

Substitute in the known values.

$\cot (x) = \frac{- 8}{\sqrt{17}}$

Simplify the value of $\cot (x)$ .

Move the negative in front of the fraction.

$\cot (x) = - \frac{8}{\sqrt{17}}$

Multiply $\frac{8}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\cot (x) = - (\frac{8}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}})$

Combine and simplify the denominator.

Multiply $\frac{8}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\cot (x) = - \frac{8 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Raise $\sqrt{17}$ to the power of $1$ .

$\cot (x) = - \frac{8 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Raise $\sqrt{17}$ to the power of $1$ .

$\cot (x) = - \frac{8 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cot (x) = - \frac{8 \sqrt{17}}{{\sqrt{17}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (x) = - \frac{8 \sqrt{17}}{{\sqrt{17}}^{2}}$

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\cot (x) = - \frac{8 \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cot (x) = - \frac{8 \sqrt{17}}{17^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\cot (x) = - \frac{8 \sqrt{17}}{17^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cot (x) = - \frac{8 \sqrt{17}}{17^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (x) = - \frac{8 \sqrt{17}}{17}$

Evaluate the exponent.

$\cot (x) = - \frac{8 \sqrt{17}}{17}$

Step 8

Find the value of secant.

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

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

Substitute in the known values.

$\sec (x) = \frac{9}{- 8}$

Move the negative in front of the fraction.

$\sec (x) = - \frac{9}{8}$

Step 9

Find the value of cosecant.

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

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

Substitute in the known values.

$\csc (x) = \frac{9}{\sqrt{17}}$

Simplify the value of $\csc (x)$ .

Multiply $\frac{9}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\csc (x) = \frac{9}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}}$

Combine and simplify the denominator.

Multiply $\frac{9}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$\csc (x) = \frac{9 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Raise $\sqrt{17}$ to the power of $1$ .

$\csc (x) = \frac{9 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Raise $\sqrt{17}$ to the power of $1$ .

$\csc (x) = \frac{9 \sqrt{17}}{\sqrt{17} \sqrt{17}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\csc (x) = \frac{9 \sqrt{17}}{{\sqrt{17}}^{1 + 1}}$

Add $1$ and $1$ .

$\csc (x) = \frac{9 \sqrt{17}}{{\sqrt{17}}^{2}}$

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$\csc (x) = \frac{9 \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\csc (x) = \frac{9 \sqrt{17}}{17^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\csc (x) = \frac{9 \sqrt{17}}{17^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\csc (x) = \frac{9 \sqrt{17}}{17^{\frac{2}{2}}}$

Rewrite the expression.

$\csc (x) = \frac{9 \sqrt{17}}{17}$

Evaluate the exponent.

$\csc (x) = \frac{9 \sqrt{17}}{17}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{\sqrt{17}}{9}$

$\cos (x) = - \frac{8}{9}$

$\tan (x) = - \frac{\sqrt{17}}{8}$

$\cot (x) = - \frac{8 \sqrt{17}}{17}$

$\sec (x) = - \frac{9}{8}$

$\csc (x) = \frac{9 \sqrt{17}}{17}$

Find the Other Trig Values in Quadrant II cos(x)=-8/9

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