Find the Other Trig Values in Quadrant II cos(theta)=-7/8

$\cos (θ) = - \frac{7}{8}$

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

$\cos (θ) = \frac{adjacent}{hypotenuse}$

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.

$Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Replace the known values in the equation.

$Opposite = \sqrt{{(8)}^{2} - {(- 7)}^{2}}$

Simplify inside the radical.

Raise $8$ to the power of $2$ .

Opposite $= \sqrt{64 - {(- 7)}^{2}}$

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

Opposite $= \sqrt{64 - 1 \cdot 49}$

Multiply $- 1$ by $49$ .

Opposite $= \sqrt{64 - 49}$

Subtract $49$ from $64$ .

Opposite $= \sqrt{15}$

Find the value of sine.

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

$\sin (θ) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (θ) = \frac{\sqrt{15}}{8}$

Find the value of tangent.

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

$\tan (θ) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (θ) = \frac{\sqrt{15}}{- 7}$

Move the negative in front of the fraction.

$\tan (θ) = - \frac{\sqrt{15}}{7}$

Find the value of cotangent.

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

$\cot (θ) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (θ) = \frac{- 7}{\sqrt{15}}$

Simplify the value of $\cot (θ)$ .

Move the negative in front of the fraction.

$\cot (θ) = - \frac{7}{\sqrt{15}}$

Multiply $\frac{7}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\cot (θ) = - (\frac{7}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}})$

Combine and simplify the denominator.

Multiply $\frac{7}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\cot (θ) = - \frac{7 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

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

$\cot (θ) = - \frac{7 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

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

$\cot (θ) = - \frac{7 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

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

$\cot (θ) = - \frac{7 \sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (θ) = - \frac{7 \sqrt{15}}{{\sqrt{15}}^{2}}$

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

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

$\cot (θ) = - \frac{7 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

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

$\cot (θ) = - \frac{7 \sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

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

$\cot (θ) = - \frac{7 \sqrt{15}}{15^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cot (θ) = - \frac{7 \sqrt{15}}{15^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (θ) = - \frac{7 \sqrt{15}}{15}$

Evaluate the exponent.

$\cot (θ) = - \frac{7 \sqrt{15}}{15}$

Find the value of secant.

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

$\sec (θ) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (θ) = \frac{8}{- 7}$

Move the negative in front of the fraction.

$\sec (θ) = - \frac{8}{7}$

Find the value of cosecant.

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

$\csc (θ) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (θ) = \frac{8}{\sqrt{15}}$

Simplify the value of $\csc (θ)$ .

Multiply $\frac{8}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\csc (θ) = \frac{8}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}$

Combine and simplify the denominator.

Multiply $\frac{8}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\csc (θ) = \frac{8 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

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

$\csc (θ) = \frac{8 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

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

$\csc (θ) = \frac{8 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

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

$\csc (θ) = \frac{8 \sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Add $1$ and $1$ .

$\csc (θ) = \frac{8 \sqrt{15}}{{\sqrt{15}}^{2}}$

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

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

$\csc (θ) = \frac{8 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

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

$\csc (θ) = \frac{8 \sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

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

$\csc (θ) = \frac{8 \sqrt{15}}{15^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\csc (θ) = \frac{8 \sqrt{15}}{15^{\frac{2}{2}}}$

Rewrite the expression.

$\csc (θ) = \frac{8 \sqrt{15}}{15}$

Evaluate the exponent.

$\csc (θ) = \frac{8 \sqrt{15}}{15}$

This is the solution to each trig value.

$\sin (θ) = \frac{\sqrt{15}}{8}$

$\cos (θ) = - \frac{7}{8}$

$\tan (θ) = - \frac{\sqrt{15}}{7}$

$\cot (θ) = - \frac{7 \sqrt{15}}{15}$

$\sec (θ) = - \frac{8}{7}$

$\csc (θ) = \frac{8 \sqrt{15}}{15}$

Find the Other Trig Values in Quadrant II cos(theta)=-7/8

Related Questions

Support LCMGCF.com by disabling your adblocker.