Find the Trig Value sin(x)=6/11

$\sin (x) = \frac{6}{11}$

Step 1

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

$\sin (x) = \frac{opposite}{hypotenuse}$

Step 2

Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.

$Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = \sqrt{{(11)}^{2} - {(6)}^{2}}$

Step 4

Simplify inside the radical.

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

Adjacent $= \sqrt{121 - {(6)}^{2}}$

Raise $6$ to the power of $2$ .

Adjacent $= \sqrt{121 - 1 \cdot 36}$

Multiply $- 1$ by $36$ .

Adjacent $= \sqrt{121 - 36}$

Subtract $36$ from $121$ .

Adjacent $= \sqrt{85}$

Step 5

Find the value of cosine.

Use the definition of cosine to find the value of $\cos (x)$ .

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

Substitute in the known values.

$\cos (x) = \frac{\sqrt{85}}{11}$

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.

$\tan (x) = \frac{6}{\sqrt{85}}$

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

Multiply $\frac{6}{\sqrt{85}}$ by $\frac{\sqrt{85}}{\sqrt{85}}$ .

$\tan (x) = \frac{6}{\sqrt{85}} \cdot \frac{\sqrt{85}}{\sqrt{85}}$

Combine and simplify the denominator.

Multiply $\frac{6}{\sqrt{85}}$ by $\frac{\sqrt{85}}{\sqrt{85}}$ .

$\tan (x) = \frac{6 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Raise $\sqrt{85}$ to the power of $1$ .

$\tan (x) = \frac{6 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Raise $\sqrt{85}$ to the power of $1$ .

$\tan (x) = \frac{6 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

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

$\tan (x) = \frac{6 \sqrt{85}}{{\sqrt{85}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (x) = \frac{6 \sqrt{85}}{{\sqrt{85}}^{2}}$

Rewrite ${\sqrt{85}}^{2}$ as $85$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{85}$ as $85^{\frac{1}{2}}$ .

$\tan (x) = \frac{6 \sqrt{85}}{{(85^{\frac{1}{2}})}^{2}}$

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

$\tan (x) = \frac{6 \sqrt{85}}{85^{\frac{1}{2} \cdot 2}}$

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

$\tan (x) = \frac{6 \sqrt{85}}{85^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (x) = \frac{6 \sqrt{85}}{85^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (x) = \frac{6 \sqrt{85}}{85}$

Evaluate the exponent.

$\tan (x) = \frac{6 \sqrt{85}}{85}$

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.

$\cot (x) = \frac{\sqrt{85}}{6}$

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.

$\sec (x) = \frac{11}{\sqrt{85}}$

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

Multiply $\frac{11}{\sqrt{85}}$ by $\frac{\sqrt{85}}{\sqrt{85}}$ .

$\sec (x) = \frac{11}{\sqrt{85}} \cdot \frac{\sqrt{85}}{\sqrt{85}}$

Combine and simplify the denominator.

Multiply $\frac{11}{\sqrt{85}}$ by $\frac{\sqrt{85}}{\sqrt{85}}$ .

$\sec (x) = \frac{11 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Raise $\sqrt{85}$ to the power of $1$ .

$\sec (x) = \frac{11 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

Raise $\sqrt{85}$ to the power of $1$ .

$\sec (x) = \frac{11 \sqrt{85}}{\sqrt{85} \sqrt{85}}$

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

$\sec (x) = \frac{11 \sqrt{85}}{{\sqrt{85}}^{1 + 1}}$

Add $1$ and $1$ .

$\sec (x) = \frac{11 \sqrt{85}}{{\sqrt{85}}^{2}}$

Rewrite ${\sqrt{85}}^{2}$ as $85$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{85}$ as $85^{\frac{1}{2}}$ .

$\sec (x) = \frac{11 \sqrt{85}}{{(85^{\frac{1}{2}})}^{2}}$

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

$\sec (x) = \frac{11 \sqrt{85}}{85^{\frac{1}{2} \cdot 2}}$

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

$\sec (x) = \frac{11 \sqrt{85}}{85^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sec (x) = \frac{11 \sqrt{85}}{85^{\frac{2}{2}}}$

Rewrite the expression.

$\sec (x) = \frac{11 \sqrt{85}}{85}$

Evaluate the exponent.

$\sec (x) = \frac{11 \sqrt{85}}{85}$

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.

$\csc (x) = \frac{11}{6}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{6}{11}$

$\cos (x) = \frac{\sqrt{85}}{11}$

$\tan (x) = \frac{6 \sqrt{85}}{85}$

$\cot (x) = \frac{\sqrt{85}}{6}$

$\sec (x) = \frac{11 \sqrt{85}}{85}$

$\csc (x) = \frac{11}{6}$

Find the Trig Value sin(x)=6/11

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