Find the Other Trig Values in Quadrant III cos(x)=-4/9

$\cos (x) = - \frac{4}{9}$

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.

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

Step 4

Simplify inside the radical.

Negate $\sqrt{{(9)}^{2} - {(- 4)}^{2}}$ .

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

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

Opposite $= - \sqrt{81 - {(- 4)}^{2}}$

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

Opposite $= - \sqrt{81 - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

Opposite $= - \sqrt{81 - 16}$

Subtract $16$ from $81$ .

Opposite $= - \sqrt{65}$

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.

$\sin (x) = \frac{- \sqrt{65}}{9}$

Move the negative in front of the fraction.

$\sin (x) = - \frac{\sqrt{65}}{9}$

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.

$\tan (x) = \frac{- \sqrt{65}}{- 4}$

Dividing two negative values results in a positive value.

$\tan (x) = \frac{\sqrt{65}}{4}$

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.

$\cot (x) = \frac{- 4}{- \sqrt{65}}$

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

Dividing two negative values results in a positive value.

$\cot (x) = \frac{4}{\sqrt{65}}$

Multiply $\frac{4}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\cot (x) = \frac{4}{\sqrt{65}} \cdot \frac{\sqrt{65}}{\sqrt{65}}$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\cot (x) = \frac{4 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Raise $\sqrt{65}$ to the power of $1$ .

$\cot (x) = \frac{4 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Raise $\sqrt{65}$ to the power of $1$ .

$\cot (x) = \frac{4 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

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

$\cot (x) = \frac{4 \sqrt{65}}{{\sqrt{65}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (x) = \frac{4 \sqrt{65}}{{\sqrt{65}}^{2}}$

Rewrite ${\sqrt{65}}^{2}$ as $65$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{65}$ as $65^{\frac{1}{2}}$ .

$\cot (x) = \frac{4 \sqrt{65}}{{(65^{\frac{1}{2}})}^{2}}$

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

$\cot (x) = \frac{4 \sqrt{65}}{65^{\frac{1}{2} \cdot 2}}$

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

$\cot (x) = \frac{4 \sqrt{65}}{65^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cot (x) = \frac{4 \sqrt{65}}{65^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (x) = \frac{4 \sqrt{65}}{65}$

Evaluate the exponent.

$\cot (x) = \frac{4 \sqrt{65}}{65}$

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.

$\sec (x) = \frac{9}{- 4}$

Move the negative in front of the fraction.

$\sec (x) = - \frac{9}{4}$

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.

$\csc (x) = \frac{9}{- \sqrt{65}}$

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

Move the negative in front of the fraction.

$\csc (x) = - \frac{9}{\sqrt{65}}$

Multiply $\frac{9}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\csc (x) = - (\frac{9}{\sqrt{65}} \cdot \frac{\sqrt{65}}{\sqrt{65}})$

Combine and simplify the denominator.

Multiply $\frac{9}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\csc (x) = - \frac{9 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Raise $\sqrt{65}$ to the power of $1$ .

$\csc (x) = - \frac{9 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Raise $\sqrt{65}$ to the power of $1$ .

$\csc (x) = - \frac{9 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

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

$\csc (x) = - \frac{9 \sqrt{65}}{{\sqrt{65}}^{1 + 1}}$

Add $1$ and $1$ .

$\csc (x) = - \frac{9 \sqrt{65}}{{\sqrt{65}}^{2}}$

Rewrite ${\sqrt{65}}^{2}$ as $65$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{65}$ as $65^{\frac{1}{2}}$ .

$\csc (x) = - \frac{9 \sqrt{65}}{{(65^{\frac{1}{2}})}^{2}}$

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

$\csc (x) = - \frac{9 \sqrt{65}}{65^{\frac{1}{2} \cdot 2}}$

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

$\csc (x) = - \frac{9 \sqrt{65}}{65^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\csc (x) = - \frac{9 \sqrt{65}}{65^{\frac{2}{2}}}$

Rewrite the expression.

$\csc (x) = - \frac{9 \sqrt{65}}{65}$

Evaluate the exponent.

$\csc (x) = - \frac{9 \sqrt{65}}{65}$

Step 10

This is the solution to each trig value.

$\sin (x) = - \frac{\sqrt{65}}{9}$

$\cos (x) = - \frac{4}{9}$

$\tan (x) = \frac{\sqrt{65}}{4}$

$\cot (x) = \frac{4 \sqrt{65}}{65}$

$\sec (x) = - \frac{9}{4}$

$\csc (x) = - \frac{9 \sqrt{65}}{65}$

Find the Other Trig Values in Quadrant III cos(x)=-4/9

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