Find the Other Trig Values in Quadrant II sec(t)=-4/3

$\sec (t) = \frac{- 4}{3}$

Step 1

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

$\sec (t) = \frac{hypotenuse}{adjacent}$

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.

$Opposite = \sqrt{{(- 4)}^{2} - {(3)}^{2}}$

Step 4

Simplify inside the radical.

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

Opposite $= \sqrt{16 - {(3)}^{2}}$

Raise $3$ to the power of $2$ .

Opposite $= \sqrt{16 - 1 \cdot 9}$

Multiply $- 1$ by $9$ .

Opposite $= \sqrt{16 - 9}$

Subtract $9$ from $16$ .

Opposite $= \sqrt{7}$

Step 5

Find the value of sine.

Use the definition of sine to find the value of $\sin (t)$ .

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

Substitute in the known values.

$\sin (t) = \frac{\sqrt{7}}{- 4}$

Move the negative in front of the fraction.

$\sin (t) = - \frac{\sqrt{7}}{4}$

Step 6

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.

$\cos (t) = \frac{3}{- 4}$

Move the negative in front of the fraction.

$\cos (t) = - \frac{3}{4}$

Step 7

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.

$\tan (t) = \frac{\sqrt{7}}{3}$

Step 8

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.

$\cot (t) = \frac{3}{\sqrt{7}}$

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

Multiply $\frac{3}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cot (t) = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Combine and simplify the denominator.

Multiply $\frac{3}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cot (t) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$\cot (t) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$\cot (t) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

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

$\cot (t) = \frac{3 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (t) = \frac{3 \sqrt{7}}{{\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\cot (t) = \frac{3 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

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

$\cot (t) = \frac{3 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

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

$\cot (t) = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cot (t) = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (t) = \frac{3 \sqrt{7}}{7}$

Evaluate the exponent.

$\cot (t) = \frac{3 \sqrt{7}}{7}$

Step 9

Move the negative in front of the fraction.

$\sec (t) = - \frac{4}{3}$

Step 10

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.

$\csc (t) = \frac{- 4}{\sqrt{7}}$

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

Move the negative in front of the fraction.

$\csc (t) = - \frac{4}{\sqrt{7}}$

Multiply $\frac{4}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\csc (t) = - (\frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\csc (t) = - \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$\csc (t) = - \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$\csc (t) = - \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

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

$\csc (t) = - \frac{4 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$\csc (t) = - \frac{4 \sqrt{7}}{{\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\csc (t) = - \frac{4 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

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

$\csc (t) = - \frac{4 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

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

$\csc (t) = - \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\csc (t) = - \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Rewrite the expression.

$\csc (t) = - \frac{4 \sqrt{7}}{7}$

Evaluate the exponent.

$\csc (t) = - \frac{4 \sqrt{7}}{7}$

Step 11

This is the solution to each trig value.

$\sin (t) = - \frac{\sqrt{7}}{4}$

$\cos (t) = - \frac{3}{4}$

$\tan (t) = \frac{\sqrt{7}}{3}$

$\cot (t) = \frac{3 \sqrt{7}}{7}$

$\sec (t) = - \frac{4}{3}$

$\csc (t) = - \frac{4 \sqrt{7}}{7}$

Find the Other Trig Values in Quadrant II sec(t)=-4/3

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