The angle $\theta_{1}$ is located in Quadrant II, and $\cos \left(\theta_{1}\right)=-\frac{22}{29}$ . $\newline$ What is the value of $\sin \left(\theta_{1}\right)$ ? Express your answer exactly. $\newline$ $\sin \left(\theta_{1}\right)=$

View step-by-step help

Home

Math Problems

Algebra 2

Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q. The angle

\theta_{1}

is located in Quadrant II, and

\cos \left(\theta_{1}\right)=-\frac{22}{29}

\newline

What is the value of

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

? Express your answer exactly.

\newline

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

Quadrant II and Pythagorean Identity: We know that in Quadrant $ext{II}$ , the sine function is positive, and we can use the Pythagorean identity $ext{sin}^2(\theta) + ext{cos}^2(\theta) = 1$ to find the value of $ext{sin}(\theta)$ .
Substituting $\cos(\theta_{1})$ into the Identity: First, we substitute the given value of $\cos(\theta_{1})$ into the Pythagorean identity. $\sin^2(\theta_{1}) + \left(-\frac{22}{29}\right)^2 = 1$
Calculating the Square of $\cos(\theta_{1})$ : Next, we calculate the square of $\cos(\theta_{1})$ . $\left(-\frac{22}{29}\right)^2 = \frac{484}{841}$
Substituting the Value Back into the Identity: Now, we substitute this value back into the Pythagorean identity. $\sin^2(\theta_{1}) + \frac{484}{841} = 1$
Solving for $\sin^2(\theta_{1})$ : We then solve for $\sin^2(\theta_{1})$ . $\sin^2(\theta_{1}) = 1 - \frac{484}{841}$
Subtracting the Fraction from $1$ : Subtracting the fraction from $1$ gives us: $\newline$ $\sin^2(\theta_{1}) = \frac{841}{841} - \frac{484}{841}$
Simplifying the Subtraction: Simplifying the subtraction, we get: $\sin^2(\theta_{1}) = \frac{841 - 484}{841}$
Performing the Subtraction in the Numerator: Performing the subtraction in the numerator gives us: $\sin^2(\theta_{1}) = \frac{357}{841}$
Taking the Positive Square Root: Since we are looking for $\sin(\theta_{1})$ and we know it should be positive in Quadrant II, we take the positive square root of the result. $\newline$ $\sin(\theta_{1}) = \sqrt{\frac{357}{841}}$
Simplifying the Square Root: Simplifying the square root, we find that $841$ is a perfect square, being $29^2$ , so: $\newline$ $\sin(\theta_{1}) = \frac{\sqrt{357}}{29}$