The angle theta_(1) is located in Quadrant IV, and sin(theta_(1))=-(24)/(25). 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)) = -(24/25). Use the Pythagorean identity sin²(theta) + cos²(theta) = 1 to find the value of cos(theta_(1)). Substitute -(24/25) for sin(theta_(1)) in sin²(theta_(1)) + cos²(theta_(1)) = 1. (-24/25)² + cos²(theta_(1)) = 1.Step 2: Substitute sin(theta_(1)) in Pythagorean identity: Simplify (-24/25)² + cos²(theta_(1)) = 1 to find the value of cos²(theta_(1)). (24/25)² + cos²(theta_(1)) = 1 (576/625) + cos²(theta_(1)) = 1 cos²(theta_(1)) = 1 - (576/625) cos²(theta_(1)) = (625/625) - (576/625) cos²(theta_(1)) = 49/625Step 3: Simplify the equation: Find the square root of cos²(theta_(1)) to get cos(theta_(1)). cos(theta_(1)) = ±√(49/625) cos(theta_(1)) = ±(7/25)Step 4: Calculate 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 IV, cos(theta_(1)) is positive. cos(theta_(1)) = 7/25

Home

Math Problems

Precalculus

Find trigonometric ratios using multiple identities

Full solution

Q. The angle

\theta_{1}

is located in Quadrant IV, and

\sin \left(\theta_{1}\right)=-\frac{24}{25}

\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 $\sin(\theta_{1}) = -\frac{24}{25}$ . Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ to find the value of $\cos(\theta_{1})$ . $\newline$ Substitute $-\frac{24}{25}$ for $\sin(\theta_{1})$ in $\sin^2(\theta_{1}) + \cos^2(\theta_{1}) = 1$ . $\newline$ $\left(-\frac{24}{25}\right)^2 + \cos^2(\theta_{1}) = 1$ .
Step $2$ : Substitute $\sin(\theta_{1})$ in Pythagorean identity: Simplify $(-\frac{24}{25})^2 + \cos^2(\theta_{1}) = 1$ to find the value of $\cos^2(\theta_{1})$ .
$\left(\frac{24}{25}\right)^2 + \cos^2(\theta_{1}) = 1$
$\left(\frac{576}{625}\right) + \cos^2(\theta_{1}) = 1$
$\cos^2(\theta_{1}) = 1 - \left(\frac{576}{625}\right)$
$\cos^2(\theta_{1}) = \left(\frac{625}{625}\right) - \left(\frac{576}{625}\right)$
$\cos^2(\theta_{1}) = \frac{49}{625}$
Step $3$ : Simplify the equation: Find the square root of $\cos^2(\theta_{1})$ to get $\cos(\theta_{1})$ . $\cos(\theta_{1}) = \pm\sqrt{\frac{49}{625}}$ $\cos(\theta_{1}) = \pm\frac{7}{25}$
Step $4$ : Calculate $\cos(\theta_{1})$ : Determine the sign of $\cos(\theta_{1})$ based on the quadrant in which $\theta_{1}$ is located. $\newline$ Since $\theta_{1}$ is in Quadrant IV, $\cos(\theta_{1})$ is positive. $\newline$ $\cos(\theta_{1}) = \frac{7}{25}$