Find the Other Trig Values in Quadrant III sin(x)=-4/7

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

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.

$Adjacent = - \sqrt{{(7)}^{2} - {(- 4)}^{2}}$

Step 4

Simplify inside the radical.

Negate $\sqrt{{(7)}^{2} - {(- 4)}^{2}}$ .

Adjacent $= - \sqrt{{(7)}^{2} - {(- 4)}^{2}}$

Raise $7$ to the power of $2$ .

Adjacent $= - \sqrt{49 - {(- 4)}^{2}}$

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

Adjacent $= - \sqrt{49 - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

Adjacent $= - \sqrt{49 - 16}$

Subtract $16$ from $49$ .

Adjacent $= - \sqrt{33}$

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.

$\cos (x) = \frac{- \sqrt{33}}{7}$

Move the negative in front of the fraction.

$\cos (x) = - \frac{\sqrt{33}}{7}$

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.

$\tan (x) = \frac{- 4}{- \sqrt{33}}$

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

Dividing two negative values results in a positive value.

$\tan (x) = \frac{4}{\sqrt{33}}$

Multiply $\frac{4}{\sqrt{33}}$ by $\frac{\sqrt{33}}{\sqrt{33}}$ .

$\tan (x) = \frac{4}{\sqrt{33}} \cdot \frac{\sqrt{33}}{\sqrt{33}}$

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{33}}$ by $\frac{\sqrt{33}}{\sqrt{33}}$ .

$\tan (x) = \frac{4 \sqrt{33}}{\sqrt{33} \sqrt{33}}$

Raise $\sqrt{33}$ to the power of $1$ .

$\tan (x) = \frac{4 \sqrt{33}}{\sqrt{33} \sqrt{33}}$

Raise $\sqrt{33}$ to the power of $1$ .

$\tan (x) = \frac{4 \sqrt{33}}{\sqrt{33} \sqrt{33}}$

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

$\tan (x) = \frac{4 \sqrt{33}}{{\sqrt{33}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (x) = \frac{4 \sqrt{33}}{{\sqrt{33}}^{2}}$

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$\tan (x) = \frac{4 \sqrt{33}}{{(33^{\frac{1}{2}})}^{2}}$

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

$\tan (x) = \frac{4 \sqrt{33}}{33^{\frac{1}{2} \cdot 2}}$

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

$\tan (x) = \frac{4 \sqrt{33}}{33^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (x) = \frac{4 \sqrt{33}}{33^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (x) = \frac{4 \sqrt{33}}{33}$

Evaluate the exponent.

$\tan (x) = \frac{4 \sqrt{33}}{33}$

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.

$\cot (x) = \frac{- \sqrt{33}}{- 4}$

Dividing two negative values results in a positive value.

$\cot (x) = \frac{\sqrt{33}}{4}$

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.

$\sec (x) = \frac{7}{- \sqrt{33}}$

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

Move the negative in front of the fraction.

$\sec (x) = - \frac{7}{\sqrt{33}}$

Multiply $\frac{7}{\sqrt{33}}$ by $\frac{\sqrt{33}}{\sqrt{33}}$ .

$\sec (x) = - (\frac{7}{\sqrt{33}} \cdot \frac{\sqrt{33}}{\sqrt{33}})$

Combine and simplify the denominator.

Multiply $\frac{7}{\sqrt{33}}$ by $\frac{\sqrt{33}}{\sqrt{33}}$ .

$\sec (x) = - \frac{7 \sqrt{33}}{\sqrt{33} \sqrt{33}}$

Raise $\sqrt{33}$ to the power of $1$ .

$\sec (x) = - \frac{7 \sqrt{33}}{\sqrt{33} \sqrt{33}}$

Raise $\sqrt{33}$ to the power of $1$ .

$\sec (x) = - \frac{7 \sqrt{33}}{\sqrt{33} \sqrt{33}}$

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

$\sec (x) = - \frac{7 \sqrt{33}}{{\sqrt{33}}^{1 + 1}}$

Add $1$ and $1$ .

$\sec (x) = - \frac{7 \sqrt{33}}{{\sqrt{33}}^{2}}$

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$\sec (x) = - \frac{7 \sqrt{33}}{{(33^{\frac{1}{2}})}^{2}}$

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

$\sec (x) = - \frac{7 \sqrt{33}}{33^{\frac{1}{2} \cdot 2}}$

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

$\sec (x) = - \frac{7 \sqrt{33}}{33^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sec (x) = - \frac{7 \sqrt{33}}{33^{\frac{2}{2}}}$

Rewrite the expression.

$\sec (x) = - \frac{7 \sqrt{33}}{33}$

Evaluate the exponent.

$\sec (x) = - \frac{7 \sqrt{33}}{33}$

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.

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

Move the negative in front of the fraction.

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

Step 10

This is the solution to each trig value.

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

$\cos (x) = - \frac{\sqrt{33}}{7}$

$\tan (x) = \frac{4 \sqrt{33}}{33}$

$\cot (x) = \frac{\sqrt{33}}{4}$

$\sec (x) = - \frac{7 \sqrt{33}}{33}$

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

Find the Other Trig Values in Quadrant III sin(x)=-4/7

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