Find the Other Trig Values in Quadrant III sin(t)=-1/4

$\sin (t) = - \frac{1}{4}$

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.

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

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.

$Adjacent = - \sqrt{{(4)}^{2} - {(- 1)}^{2}}$

Step 4

Simplify inside the radical.

Negate $\sqrt{{(4)}^{2} - {(- 1)}^{2}}$ .

Adjacent $= - \sqrt{{(4)}^{2} - {(- 1)}^{2}}$

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

Adjacent $= - \sqrt{16 - {(- 1)}^{2}}$

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Multiply $- 1$ by ${(- 1)}^{2}$ .

Raise $- 1$ to the power of $1$ .

Adjacent $= - \sqrt{16 + (- 1) {(- 1)}^{2}}$

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

Adjacent $= - \sqrt{16 + {(- 1)}^{1 + 2}}$

Add $1$ and $2$ .

Adjacent $= - \sqrt{16 + {(- 1)}^{3}}$

Raise $- 1$ to the power of $3$ .

Adjacent $= - \sqrt{16 - 1}$

Subtract $1$ from $16$ .

Adjacent $= - \sqrt{15}$

Step 5

Find the value of cosine.

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

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

Substitute in the known values.

$\cos (t) = \frac{- \sqrt{15}}{4}$

Move the negative in front of the fraction.

$\cos (t) = - \frac{\sqrt{15}}{4}$

Step 6

Find the value of tangent.

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

$\tan (t) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (t) = \frac{- 1}{- \sqrt{15}}$

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

Dividing two negative values results in a positive value.

$\tan (t) = \frac{1}{\sqrt{15}}$

Multiply $\frac{1}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\tan (t) = \frac{1}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\tan (t) = \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

Raise $\sqrt{15}$ to the power of $1$ .

$\tan (t) = \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

Raise $\sqrt{15}$ to the power of $1$ .

$\tan (t) = \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

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

$\tan (t) = \frac{\sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (t) = \frac{\sqrt{15}}{{\sqrt{15}}^{2}}$

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\tan (t) = \frac{\sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

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

$\tan (t) = \frac{\sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

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

$\tan (t) = \frac{\sqrt{15}}{15^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (t) = \frac{\sqrt{15}}{15^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (t) = \frac{\sqrt{15}}{15}$

Evaluate the exponent.

$\tan (t) = \frac{\sqrt{15}}{15}$

Step 7

Find the value of cotangent.

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

$\cot (t) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (t) = \frac{- \sqrt{15}}{- 1}$

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

Dividing two negative values results in a positive value.

$\cot (t) = \frac{\sqrt{15}}{1}$

Divide $\sqrt{15}$ by $1$ .

$\cot (t) = \sqrt{15}$

Step 8

Find the value of secant.

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

$\sec (t) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (t) = \frac{4}{- \sqrt{15}}$

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

Move the negative in front of the fraction.

$\sec (t) = - \frac{4}{\sqrt{15}}$

Multiply $\frac{4}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\sec (t) = - (\frac{4}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}})$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\sec (t) = - \frac{4 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Raise $\sqrt{15}$ to the power of $1$ .

$\sec (t) = - \frac{4 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Raise $\sqrt{15}$ to the power of $1$ .

$\sec (t) = - \frac{4 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

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

$\sec (t) = - \frac{4 \sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Add $1$ and $1$ .

$\sec (t) = - \frac{4 \sqrt{15}}{{\sqrt{15}}^{2}}$

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\sec (t) = - \frac{4 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

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

$\sec (t) = - \frac{4 \sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

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

$\sec (t) = - \frac{4 \sqrt{15}}{15^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sec (t) = - \frac{4 \sqrt{15}}{15^{\frac{2}{2}}}$

Rewrite the expression.

$\sec (t) = - \frac{4 \sqrt{15}}{15}$

Evaluate the exponent.

$\sec (t) = - \frac{4 \sqrt{15}}{15}$

Step 9

Find the value of cosecant.

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

$\csc (t) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (t) = \frac{4}{- 1}$

Divide $4$ by $- 1$ .

$\csc (t) = - 4$

Step 10

This is the solution to each trig value.

$\sin (t) = - \frac{1}{4}$

$\cos (t) = - \frac{\sqrt{15}}{4}$

$\tan (t) = \frac{\sqrt{15}}{15}$

$\cot (t) = \sqrt{15}$

$\sec (t) = - \frac{4 \sqrt{15}}{15}$

$\csc (t) = - 4$

Find the Other Trig Values in Quadrant III sin(t)=-1/4

Related Questions

Support LCMGCF.com by disabling your adblocker.