Find the Other Trig Values in Quadrant III sin(0)=-7/10

$\sin (0) = - \frac{7}{10}$

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.

$\sin (0) = \frac{opposite}{hypotenuse}$

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.

$Adjacent = - \sqrt{{(10)}^{2} - {(- 7)}^{2}}$

Step 4

Simplify inside the radical.

Negate $\sqrt{{(10)}^{2} - {(- 7)}^{2}}$ .

Adjacent $= - \sqrt{{(10)}^{2} - {(- 7)}^{2}}$

Raise $10$ to the power of $2$ .

Adjacent $= - \sqrt{100 - {(- 7)}^{2}}$

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

Adjacent $= - \sqrt{100 - 1 \cdot 49}$

Multiply $- 1$ by $49$ .

Adjacent $= - \sqrt{100 - 49}$

Subtract $49$ from $100$ .

Adjacent $= - \sqrt{51}$

Step 5

Find the value of cosine.

Use the definition of cosine to find the value of $\cos (0)$ .

$\cos (0) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (0) = \frac{- \sqrt{51}}{10}$

Move the negative in front of the fraction.

$\cos (0) = - \frac{\sqrt{51}}{10}$

Step 6

Find the value of tangent.

Use the definition of tangent to find the value of $\tan (0)$ .

$\tan (0) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (0) = \frac{- 7}{- \sqrt{51}}$

Simplify the value of $\tan (0)$ .

Dividing two negative values results in a positive value.

$\tan (0) = \frac{7}{\sqrt{51}}$

Multiply $\frac{7}{\sqrt{51}}$ by $\frac{\sqrt{51}}{\sqrt{51}}$ .

$\tan (0) = \frac{7}{\sqrt{51}} \cdot \frac{\sqrt{51}}{\sqrt{51}}$

Combine and simplify the denominator.

Multiply $\frac{7}{\sqrt{51}}$ by $\frac{\sqrt{51}}{\sqrt{51}}$ .

$\tan (0) = \frac{7 \sqrt{51}}{\sqrt{51} \sqrt{51}}$

Raise $\sqrt{51}$ to the power of $1$ .

$\tan (0) = \frac{7 \sqrt{51}}{\sqrt{51} \sqrt{51}}$

Raise $\sqrt{51}$ to the power of $1$ .

$\tan (0) = \frac{7 \sqrt{51}}{\sqrt{51} \sqrt{51}}$

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

$\tan (0) = \frac{7 \sqrt{51}}{{\sqrt{51}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (0) = \frac{7 \sqrt{51}}{{\sqrt{51}}^{2}}$

Rewrite ${\sqrt{51}}^{2}$ as $51$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{51}$ as $51^{\frac{1}{2}}$ .

$\tan (0) = \frac{7 \sqrt{51}}{{(51^{\frac{1}{2}})}^{2}}$

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

$\tan (0) = \frac{7 \sqrt{51}}{51^{\frac{1}{2} \cdot 2}}$

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

$\tan (0) = \frac{7 \sqrt{51}}{51^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (0) = \frac{7 \sqrt{51}}{51^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (0) = \frac{7 \sqrt{51}}{51}$

Evaluate the exponent.

$\tan (0) = \frac{7 \sqrt{51}}{51}$

Step 7

Find the value of cotangent.

Use the definition of cotangent to find the value of $\cot (0)$ .

$\cot (0) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (0) = \frac{- \sqrt{51}}{- 7}$

Dividing two negative values results in a positive value.

$\cot (0) = \frac{\sqrt{51}}{7}$

Step 8

Find the value of secant.

Use the definition of secant to find the value of $\sec (0)$ .

$\sec (0) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (0) = \frac{10}{- \sqrt{51}}$

Simplify the value of $\sec (0)$ .

Move the negative in front of the fraction.

$\sec (0) = - \frac{10}{\sqrt{51}}$

Multiply $\frac{10}{\sqrt{51}}$ by $\frac{\sqrt{51}}{\sqrt{51}}$ .

$\sec (0) = - (\frac{10}{\sqrt{51}} \cdot \frac{\sqrt{51}}{\sqrt{51}})$

Combine and simplify the denominator.

Multiply $\frac{10}{\sqrt{51}}$ by $\frac{\sqrt{51}}{\sqrt{51}}$ .

$\sec (0) = - \frac{10 \sqrt{51}}{\sqrt{51} \sqrt{51}}$

Raise $\sqrt{51}$ to the power of $1$ .

$\sec (0) = - \frac{10 \sqrt{51}}{\sqrt{51} \sqrt{51}}$

Raise $\sqrt{51}$ to the power of $1$ .

$\sec (0) = - \frac{10 \sqrt{51}}{\sqrt{51} \sqrt{51}}$

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

$\sec (0) = - \frac{10 \sqrt{51}}{{\sqrt{51}}^{1 + 1}}$

Add $1$ and $1$ .

$\sec (0) = - \frac{10 \sqrt{51}}{{\sqrt{51}}^{2}}$

Rewrite ${\sqrt{51}}^{2}$ as $51$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{51}$ as $51^{\frac{1}{2}}$ .

$\sec (0) = - \frac{10 \sqrt{51}}{{(51^{\frac{1}{2}})}^{2}}$

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

$\sec (0) = - \frac{10 \sqrt{51}}{51^{\frac{1}{2} \cdot 2}}$

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

$\sec (0) = - \frac{10 \sqrt{51}}{51^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sec (0) = - \frac{10 \sqrt{51}}{51^{\frac{2}{2}}}$

Rewrite the expression.

$\sec (0) = - \frac{10 \sqrt{51}}{51}$

Evaluate the exponent.

$\sec (0) = - \frac{10 \sqrt{51}}{51}$

Step 9

Find the value of cosecant.

Use the definition of cosecant to find the value of $\csc (0)$ .

$\csc (0) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (0) = \frac{10}{- 7}$

Move the negative in front of the fraction.

$\csc (0) = - \frac{10}{7}$

Step 10

This is the solution to each trig value.

$\sin (0) = - \frac{7}{10}$

$\cos (0) = - \frac{\sqrt{51}}{10}$

$\tan (0) = \frac{7 \sqrt{51}}{51}$

$\cot (0) = \frac{\sqrt{51}}{7}$

$\sec (0) = - \frac{10 \sqrt{51}}{51}$

$\csc (0) = - \frac{10}{7}$

Find the Other Trig Values in Quadrant III sin(0)=-7/10

Related Questions

Support LCMGCF.com by disabling your adblocker.