Find the Other Trig Values in Quadrant III tan(B)=15/9

$\tan (B) = \frac{15}{9}$

Step 1

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

$\tan (B) = \frac{opposite}{adjacent}$

Step 2

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

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(- 15)}^{2} + {(- 9)}^{2}}$

Step 4

Simplify inside the radical.

Raise $- 15$ to the power of $2$ .

Hypotenuse $= \sqrt{225 + {(- 9)}^{2}}$

Raise $- 9$ to the power of $2$ .

Hypotenuse $= \sqrt{225 + 81}$

Add $225$ and $81$ .

Hypotenuse $= \sqrt{306}$

Rewrite $306$ as $3^{2} \cdot 34$ .

Factor $9$ out of $306$ .

Hypotenuse $= \sqrt{9 (34)}$

Rewrite $9$ as $3^{2}$ .

Hypotenuse $= \sqrt{3^{2} \cdot 34}$

Pull terms out from under the radical.

Hypotenuse $= 3 \sqrt{34}$

Step 5

Find the value of sine.

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

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

Substitute in the known values.

$\sin (B) = \frac{- 15}{3 \sqrt{34}}$

Simplify the value of $\sin (B)$ .

Cancel the common factor of $- 15$ and $3$ .

Factor $3$ out of $- 15$ .

$\sin (B) = \frac{3 \cdot - 5}{3 \sqrt{34}}$

Cancel the common factors.

Factor $3$ out of $3 \sqrt{34}$ .

$\sin (B) = \frac{3 \cdot - 5}{3 (\sqrt{34})}$

Cancel the common factor.

$\sin (B) = \frac{3 \cdot - 5}{3 \sqrt{34}}$

Rewrite the expression.

$\sin (B) = \frac{- 5}{\sqrt{34}}$

Move the negative in front of the fraction.

$\sin (B) = - \frac{5}{\sqrt{34}}$

Multiply $\frac{5}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\sin (B) = - (\frac{5}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}})$

Combine and simplify the denominator.

Multiply $\frac{5}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\sin (B) = - \frac{5 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Raise $\sqrt{34}$ to the power of $1$ .

$\sin (B) = - \frac{5 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Raise $\sqrt{34}$ to the power of $1$ .

$\sin (B) = - \frac{5 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

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

$\sin (B) = - \frac{5 \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (B) = - \frac{5 \sqrt{34}}{{\sqrt{34}}^{2}}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\sin (B) = - \frac{5 \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

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

$\sin (B) = - \frac{5 \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

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

$\sin (B) = - \frac{5 \sqrt{34}}{34^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sin (B) = - \frac{5 \sqrt{34}}{34^{\frac{2}{2}}}$

Rewrite the expression.

$\sin (B) = - \frac{5 \sqrt{34}}{34}$

Evaluate the exponent.

$\sin (B) = - \frac{5 \sqrt{34}}{34}$

Step 6

Find the value of cosine.

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

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

Substitute in the known values.

$\cos (B) = \frac{- 9}{3 \sqrt{34}}$

Simplify the value of $\cos (B)$ .

Cancel the common factor of $- 9$ and $3$ .

Factor $3$ out of $- 9$ .

$\cos (B) = \frac{3 \cdot - 3}{3 \sqrt{34}}$

Cancel the common factors.

Factor $3$ out of $3 \sqrt{34}$ .

$\cos (B) = \frac{3 \cdot - 3}{3 (\sqrt{34})}$

Cancel the common factor.

$\cos (B) = \frac{3 \cdot - 3}{3 \sqrt{34}}$

Rewrite the expression.

$\cos (B) = \frac{- 3}{\sqrt{34}}$

Move the negative in front of the fraction.

$\cos (B) = - \frac{3}{\sqrt{34}}$

Multiply $\frac{3}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\cos (B) = - (\frac{3}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}})$

Combine and simplify the denominator.

Multiply $\frac{3}{\sqrt{34}}$ by $\frac{\sqrt{34}}{\sqrt{34}}$ .

$\cos (B) = - \frac{3 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Raise $\sqrt{34}$ to the power of $1$ .

$\cos (B) = - \frac{3 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Raise $\sqrt{34}$ to the power of $1$ .

$\cos (B) = - \frac{3 \sqrt{34}}{\sqrt{34} \sqrt{34}}$

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

$\cos (B) = - \frac{3 \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos (B) = - \frac{3 \sqrt{34}}{{\sqrt{34}}^{2}}$

Rewrite ${\sqrt{34}}^{2}$ as $34$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{34}$ as $34^{\frac{1}{2}}$ .

$\cos (B) = - \frac{3 \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

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

$\cos (B) = - \frac{3 \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

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

$\cos (B) = - \frac{3 \sqrt{34}}{34^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cos (B) = - \frac{3 \sqrt{34}}{34^{\frac{2}{2}}}$

Rewrite the expression.

$\cos (B) = - \frac{3 \sqrt{34}}{34}$

Evaluate the exponent.

$\cos (B) = - \frac{3 \sqrt{34}}{34}$

Step 7

Cancel the common factor of $15$ and $9$ .

Factor $3$ out of $15$ .

$\tan (B) = \frac{3 (5)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$\tan (B) = \frac{3 \cdot 5}{3 \cdot 3}$

Cancel the common factor.

$\tan (B) = \frac{3 \cdot 5}{3 \cdot 3}$

Rewrite the expression.

$\tan (B) = \frac{5}{3}$

Step 8

Find the value of cotangent.

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

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

Substitute in the known values.

$\cot (B) = \frac{- 9}{- 15}$

Cancel the common factor of $- 9$ and $- 15$ .

Factor $- 3$ out of $- 9$ .

$\cot (B) = \frac{- 3 \cdot 3}{- 15}$

Cancel the common factors.

Factor $- 3$ out of $- 15$ .

$\cot (B) = \frac{- 3 \cdot 3}{- 3 \cdot 5}$

Cancel the common factor.

$\cot (B) = \frac{- 3 \cdot 3}{- 3 \cdot 5}$

Rewrite the expression.

$\cot (B) = \frac{3}{5}$

Step 9

Find the value of secant.

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

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

Substitute in the known values.

$\sec (B) = \frac{3 \sqrt{34}}{- 9}$

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

Cancel the common factor of $3$ and $- 9$ .

Factor $3$ out of $3 \sqrt{34}$ .

$\sec (B) = \frac{3 (\sqrt{34})}{- 9}$

Cancel the common factors.

Factor $3$ out of $- 9$ .

$\sec (B) = \frac{3 \sqrt{34}}{3 \cdot - 3}$

Cancel the common factor.

$\sec (B) = \frac{3 \sqrt{34}}{3 \cdot - 3}$

Rewrite the expression.

$\sec (B) = \frac{\sqrt{34}}{- 3}$

Move the negative in front of the fraction.

$\sec (B) = - \frac{\sqrt{34}}{3}$

Step 10

Find the value of cosecant.

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

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

Substitute in the known values.

$\csc (B) = \frac{3 \sqrt{34}}{- 15}$

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

Cancel the common factor of $3$ and $- 15$ .

Factor $3$ out of $3 \sqrt{34}$ .

$\csc (B) = \frac{3 (\sqrt{34})}{- 15}$

Cancel the common factors.

Factor $3$ out of $- 15$ .

$\csc (B) = \frac{3 \sqrt{34}}{3 \cdot - 5}$

Cancel the common factor.

$\csc (B) = \frac{3 \sqrt{34}}{3 \cdot - 5}$

Rewrite the expression.

$\csc (B) = \frac{\sqrt{34}}{- 5}$

Move the negative in front of the fraction.

$\csc (B) = - \frac{\sqrt{34}}{5}$

Step 11

This is the solution to each trig value.

$\sin (B) = - \frac{5 \sqrt{34}}{34}$

$\cos (B) = - \frac{3 \sqrt{34}}{34}$

$\tan (B) = \frac{5}{3}$

$\cot (B) = \frac{3}{5}$

$\sec (B) = - \frac{\sqrt{34}}{3}$

$\csc (B) = - \frac{\sqrt{34}}{5}$

Find the Other Trig Values in Quadrant III tan(B)=15/9

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