Find the Other Trig Values in Quadrant II cos(theta_1)=-2/11

$\cos (θ_{1}) = - \frac{2}{11}$

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.

$\cos (θ_{1}) = \frac{adjacent}{hypotenuse}$

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.

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

Step 4

Simplify inside the radical.

Raise $11$ to the power of $2$ .

Opposite $= \sqrt{121 - {(- 2)}^{2}}$

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

Opposite $= \sqrt{121 - 1 \cdot 4}$

Multiply $- 1$ by $4$ .

Opposite $= \sqrt{121 - 4}$

Subtract $4$ from $121$ .

Opposite $= \sqrt{117}$

Rewrite $117$ as $3^{2} \cdot 13$ .

Factor $9$ out of $117$ .

Opposite $= \sqrt{9 (13)}$

Rewrite $9$ as $3^{2}$ .

Opposite $= \sqrt{3^{2} \cdot 13}$

Pull terms out from under the radical.

Opposite $= 3 \sqrt{13}$

Step 5

Find the value of sine.

Use the definition of sine to find the value of $\sin (θ_{1})$ .

$\sin (θ_{1}) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (θ_{1}) = \frac{3 \sqrt{13}}{11}$

Step 6

Find the value of tangent.

Use the definition of tangent to find the value of $\tan (θ_{1})$ .

$\tan (θ_{1}) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (θ_{1}) = \frac{3 \sqrt{13}}{- 2}$

Move the negative in front of the fraction.

$\tan (θ_{1}) = - \frac{3 \sqrt{13}}{2}$

Step 7

Find the value of cotangent.

Use the definition of cotangent to find the value of $\cot (θ_{1})$ .

$\cot (θ_{1}) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (θ_{1}) = \frac{- 2}{3 \sqrt{13}}$

Simplify the value of $\cot (θ_{1})$ .

Move the negative in front of the fraction.

$\cot (θ_{1}) = - \frac{2}{3 \sqrt{13}}$

Multiply $\frac{2}{3 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cot (θ_{1}) = - (\frac{2}{3 \sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Combine and simplify the denominator.

Multiply $\frac{2}{3 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 \sqrt{13} \sqrt{13}}$

Move $\sqrt{13}$ .

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

Raise $\sqrt{13}$ to the power of $1$ .

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

Raise $\sqrt{13}$ to the power of $1$ .

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

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

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 {\sqrt{13}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 {\sqrt{13}}^{2}}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 {(13^{\frac{1}{2}})}^{2}}$

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

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 \cdot 13^{\frac{1}{2} \cdot 2}}$

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

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 \cdot 13^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 \cdot 13^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 \cdot 13}$

Evaluate the exponent.

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{3 \cdot 13}$

Multiply $3$ by $13$ .

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{39}$

Step 8

Find the value of secant.

Use the definition of secant to find the value of $\sec (θ_{1})$ .

$\sec (θ_{1}) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (θ_{1}) = \frac{11}{- 2}$

Move the negative in front of the fraction.

$\sec (θ_{1}) = - \frac{11}{2}$

Step 9

Find the value of cosecant.

Use the definition of cosecant to find the value of $\csc (θ_{1})$ .

$\csc (θ_{1}) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (θ_{1}) = \frac{11}{3 \sqrt{13}}$

Simplify the value of $\csc (θ_{1})$ .

Multiply $\frac{11}{3 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\csc (θ_{1}) = \frac{11}{3 \sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Combine and simplify the denominator.

Multiply $\frac{11}{3 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 \sqrt{13} \sqrt{13}}$

Move $\sqrt{13}$ .

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

Raise $\sqrt{13}$ to the power of $1$ .

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

Raise $\sqrt{13}$ to the power of $1$ .

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

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

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 {\sqrt{13}}^{1 + 1}}$

Add $1$ and $1$ .

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 {\sqrt{13}}^{2}}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 {(13^{\frac{1}{2}})}^{2}}$

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

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 \cdot 13^{\frac{1}{2} \cdot 2}}$

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

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 \cdot 13^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 \cdot 13^{\frac{2}{2}}}$

Rewrite the expression.

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 \cdot 13}$

Evaluate the exponent.

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{3 \cdot 13}$

Multiply $3$ by $13$ .

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{39}$

Step 10

This is the solution to each trig value.

$\sin (θ_{1}) = \frac{3 \sqrt{13}}{11}$

$\cos (θ_{1}) = - \frac{2}{11}$

$\tan (θ_{1}) = - \frac{3 \sqrt{13}}{2}$

$\cot (θ_{1}) = - \frac{2 \sqrt{13}}{39}$

$\sec (θ_{1}) = - \frac{11}{2}$

$\csc (θ_{1}) = \frac{11 \sqrt{13}}{39}$

Find the Other Trig Values in Quadrant II cos(theta_1)=-2/11

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