The angle theta_(1) is located in Quadrant III, and sin(theta_(1))=-(sqrt3)/(2). What is the value of cos(theta_(1)) ? Express your answer exactly. cos(theta_(1))=

Step 1: Find sin(theta_(1)): We know that sin(theta_(1)) = -(sqrt3)/(2). Use the Pythagorean identity sin²(theta) + cos²(theta) = 1 to find the value of cos(theta_(1)). Substitute -(sqrt3)/(2) for sin(theta_(1)) in sin²(theta_(1)) + cos²(theta_(1)) = 1. (-sqrt3/2)² + cos²(theta_(1)) = 1.Step 2: Substitute sin(theta_(1)) in the Pythagorean identity: Simplify (-sqrt3/2)² + cos²(theta_(1)) = 1 to find the value of cos²(theta_(1)). (3/4) + cos²(theta_(1)) = 1 cos²(theta_(1)) = 1 - 3/4 cos²(theta_(1)) = 1/4Step 3: Simplify the equation: Since cos²(theta_(1)) = 1/4, we take the square root of both sides to find cos(theta_(1)). cos(theta_(1)) = ± sqrt(1/4) cos(theta_(1)) = ± 1/2Step 4: Find the value of cos(theta_(1)): Determine the sign of cos(theta_(1)) based on the quadrant in which theta_(1) is located. Since theta_(1) is in Quadrant III, both sine and cosine are negative. Therefore, cos(theta_(1)) is negative. cos(theta_(1)) = -1/2

Home

Math Problems

Precalculus

Find trigonometric ratios using multiple identities

Full solution

Q. The angle

\theta_{1}

is located in Quadrant III, and

\sin \left(\theta_{1}\right)=-\frac{\sqrt{3}}{2}

\newline

What is the value of

\cos \left(\theta_{1}\right)

? Express your answer exactly.

\newline

\cos \left(\theta_{1}\right)=

Step $1$ : Find $\sin(\theta_{1})$ : We know that $\sin(\theta_{1}) = -\frac{\sqrt{3}}{2}$ . Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ to find the value of $\cos(\theta_{1})$ . $\newline$ Substitute $-\frac{\sqrt{3}}{2}$ for $\sin(\theta_{1})$ in $\sin^2(\theta_{1}) + \cos^2(\theta_{1}) = 1$ . $\newline$ $\left(-\frac{\sqrt{3}}{2}\right)^2 + \cos^2(\theta_{1}) = 1$ .
Step $2$ : Substitute $\sin(\theta_{1})$ in the Pythagorean identity: Simplify $(-\sqrt{3}/2)^2 + \cos^2(\theta_{1}) = 1$ to find the value of $\cos^2(\theta_{1})$ . $\newline$ $(3/4) + \cos^2(\theta_{1}) = 1$ $\newline$ $\cos^2(\theta_{1}) = 1 - 3/4$ $\newline$ $\cos^2(\theta_{1}) = 1/4$
Step $3$ : Simplify the equation: Since $\cos^2(\theta_{1}) = \frac{1}{4}$ , we take the square root of both sides to find $\cos(\theta_{1})$ . $\newline$ $\cos(\theta_{1}) = \pm \sqrt{\frac{1}{4}}$ $\newline$ $\cos(\theta_{1}) = \pm \frac{1}{2}$
Step $4$ : Find the value of $\cos(\theta_{1})$ : Determine the sign of $\cos(\theta_{1})$ based on the quadrant in which $\theta_{1}$ is located. $\newline$ Since $\theta_{1}$ is in Quadrant III, both sine and cosine are negative. $\newline$ Therefore, $\cos(\theta_{1})$ is negative. $\newline$ $\cos(\theta_{1}) = -\frac{1}{2}$