Find the Other Trig Values in Quadrant II sin(theta)=2/5

$\sin (θ) = \frac{2}{5}$

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.

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

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.

$Adjacent = - \sqrt{{(5)}^{2} - {(2)}^{2}}$

Step 4

Simplify inside the radical.

Negate $\sqrt{{(5)}^{2} - {(2)}^{2}}$ .

Adjacent $= - \sqrt{{(5)}^{2} - {(2)}^{2}}$

Raise $5$ to the power of $2$ .

Adjacent $= - \sqrt{25 - {(2)}^{2}}$

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

Adjacent $= - \sqrt{25 - 1 \cdot 4}$

Multiply $- 1$ by $4$ .

Adjacent $= - \sqrt{25 - 4}$

Subtract $4$ from $25$ .

Adjacent $= - \sqrt{21}$

Step 5

Find the value of cosine.

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

$\cos (θ) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (θ) = \frac{- \sqrt{21}}{5}$

Move the negative in front of the fraction.

$\cos (θ) = - \frac{\sqrt{21}}{5}$

Step 6

Find the value of tangent.

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

$\tan (θ) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (θ) = \frac{2}{- \sqrt{21}}$

Simplify the value of $\tan (θ)$ .

Move the negative in front of the fraction.

$\tan (θ) = - \frac{2}{\sqrt{21}}$

Multiply $\frac{2}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\tan (θ) = - (\frac{2}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}})$

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Raise $\sqrt{21}$ to the power of $1$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Raise $\sqrt{21}$ to the power of $1$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

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

$\tan (θ) = - \frac{2 \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{{\sqrt{21}}^{2}}$

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\tan (θ) = - \frac{2 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

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

$\tan (θ) = - \frac{2 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

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

$\tan (θ) = - \frac{2 \sqrt{21}}{21^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (θ) = - \frac{2 \sqrt{21}}{21^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (θ) = - \frac{2 \sqrt{21}}{21}$

Evaluate the exponent.

$\tan (θ) = - \frac{2 \sqrt{21}}{21}$

Step 7

Find the value of cotangent.

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

$\cot (θ) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (θ) = \frac{- \sqrt{21}}{2}$

Move the negative in front of the fraction.

$\cot (θ) = - \frac{\sqrt{21}}{2}$

Step 8

Find the value of secant.

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

$\sec (θ) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (θ) = \frac{5}{- \sqrt{21}}$

Simplify the value of $\sec (θ)$ .

Move the negative in front of the fraction.

$\sec (θ) = - \frac{5}{\sqrt{21}}$

Multiply $\frac{5}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\sec (θ) = - (\frac{5}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}})$

Combine and simplify the denominator.

Multiply $\frac{5}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\sec (θ) = - \frac{5 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Raise $\sqrt{21}$ to the power of $1$ .

$\sec (θ) = - \frac{5 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

Raise $\sqrt{21}$ to the power of $1$ .

$\sec (θ) = - \frac{5 \sqrt{21}}{\sqrt{21} \sqrt{21}}$

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

$\sec (θ) = - \frac{5 \sqrt{21}}{{\sqrt{21}}^{1 + 1}}$

Add $1$ and $1$ .

$\sec (θ) = - \frac{5 \sqrt{21}}{{\sqrt{21}}^{2}}$

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\sec (θ) = - \frac{5 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}$

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

$\sec (θ) = - \frac{5 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}$

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

$\sec (θ) = - \frac{5 \sqrt{21}}{21^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sec (θ) = - \frac{5 \sqrt{21}}{21^{\frac{2}{2}}}$

Rewrite the expression.

$\sec (θ) = - \frac{5 \sqrt{21}}{21}$

Evaluate the exponent.

$\sec (θ) = - \frac{5 \sqrt{21}}{21}$

Step 9

Find the value of cosecant.

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

$\csc (θ) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (θ) = \frac{5}{2}$

Step 10

This is the solution to each trig value.

$\sin (θ) = \frac{2}{5}$

$\cos (θ) = - \frac{\sqrt{21}}{5}$

$\tan (θ) = - \frac{2 \sqrt{21}}{21}$

$\cot (θ) = - \frac{\sqrt{21}}{2}$

$\sec (θ) = - \frac{5 \sqrt{21}}{21}$

$\csc (θ) = \frac{5}{2}$

Find the Other Trig Values in Quadrant II sin(theta)=2/5

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