Find the Other Trig Values in Quadrant I sin(x)=17/20

$\sin (x) = \frac{17}{20}$

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.

$Adjacent = \sqrt{{(20)}^{2} - {(17)}^{2}}$

Step 4

Simplify inside the radical.

Raise $20$ to the power of $2$ .

Adjacent $= \sqrt{400 - {(17)}^{2}}$

Raise $17$ to the power of $2$ .

Adjacent $= \sqrt{400 - 1 \cdot 289}$

Multiply $- 1$ by $289$ .

Adjacent $= \sqrt{400 - 289}$

Subtract $289$ from $400$ .

Adjacent $= \sqrt{111}$

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.

$\cos (x) = \frac{\sqrt{111}}{20}$

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.

$\tan (x) = \frac{17}{\sqrt{111}}$

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

Multiply $\frac{17}{\sqrt{111}}$ by $\frac{\sqrt{111}}{\sqrt{111}}$ .

$\tan (x) = \frac{17}{\sqrt{111}} \cdot \frac{\sqrt{111}}{\sqrt{111}}$

Combine and simplify the denominator.

Multiply $\frac{17}{\sqrt{111}}$ by $\frac{\sqrt{111}}{\sqrt{111}}$ .

$\tan (x) = \frac{17 \sqrt{111}}{\sqrt{111} \sqrt{111}}$

Raise $\sqrt{111}$ to the power of $1$ .

$\tan (x) = \frac{17 \sqrt{111}}{\sqrt{111} \sqrt{111}}$

Raise $\sqrt{111}$ to the power of $1$ .

$\tan (x) = \frac{17 \sqrt{111}}{\sqrt{111} \sqrt{111}}$

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

$\tan (x) = \frac{17 \sqrt{111}}{{\sqrt{111}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (x) = \frac{17 \sqrt{111}}{{\sqrt{111}}^{2}}$

Rewrite ${\sqrt{111}}^{2}$ as $111$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{111}$ as $111^{\frac{1}{2}}$ .

$\tan (x) = \frac{17 \sqrt{111}}{{(111^{\frac{1}{2}})}^{2}}$

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

$\tan (x) = \frac{17 \sqrt{111}}{111^{\frac{1}{2} \cdot 2}}$

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

$\tan (x) = \frac{17 \sqrt{111}}{111^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (x) = \frac{17 \sqrt{111}}{111^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (x) = \frac{17 \sqrt{111}}{111}$

Evaluate the exponent.

$\tan (x) = \frac{17 \sqrt{111}}{111}$

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.

$\cot (x) = \frac{\sqrt{111}}{17}$

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.

$\sec (x) = \frac{20}{\sqrt{111}}$

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

Multiply $\frac{20}{\sqrt{111}}$ by $\frac{\sqrt{111}}{\sqrt{111}}$ .

$\sec (x) = \frac{20}{\sqrt{111}} \cdot \frac{\sqrt{111}}{\sqrt{111}}$

Combine and simplify the denominator.

Multiply $\frac{20}{\sqrt{111}}$ by $\frac{\sqrt{111}}{\sqrt{111}}$ .

$\sec (x) = \frac{20 \sqrt{111}}{\sqrt{111} \sqrt{111}}$

Raise $\sqrt{111}$ to the power of $1$ .

$\sec (x) = \frac{20 \sqrt{111}}{\sqrt{111} \sqrt{111}}$

Raise $\sqrt{111}$ to the power of $1$ .

$\sec (x) = \frac{20 \sqrt{111}}{\sqrt{111} \sqrt{111}}$

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

$\sec (x) = \frac{20 \sqrt{111}}{{\sqrt{111}}^{1 + 1}}$

Add $1$ and $1$ .

$\sec (x) = \frac{20 \sqrt{111}}{{\sqrt{111}}^{2}}$

Rewrite ${\sqrt{111}}^{2}$ as $111$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{111}$ as $111^{\frac{1}{2}}$ .

$\sec (x) = \frac{20 \sqrt{111}}{{(111^{\frac{1}{2}})}^{2}}$

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

$\sec (x) = \frac{20 \sqrt{111}}{111^{\frac{1}{2} \cdot 2}}$

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

$\sec (x) = \frac{20 \sqrt{111}}{111^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sec (x) = \frac{20 \sqrt{111}}{111^{\frac{2}{2}}}$

Rewrite the expression.

$\sec (x) = \frac{20 \sqrt{111}}{111}$

Evaluate the exponent.

$\sec (x) = \frac{20 \sqrt{111}}{111}$

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.

$\csc (x) = \frac{20}{17}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{17}{20}$

$\cos (x) = \frac{\sqrt{111}}{20}$

$\tan (x) = \frac{17 \sqrt{111}}{111}$

$\cot (x) = \frac{\sqrt{111}}{17}$

$\sec (x) = \frac{20 \sqrt{111}}{111}$

$\csc (x) = \frac{20}{17}$

Find the Other Trig Values in Quadrant I sin(x)=17/20

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