Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\cos \theta = \frac { \sqrt { 5 } } { 5 } , \quad 0

Question

Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\cos \theta = \frac { \sqrt { 5 } } { 5 } , \quad 0 < \theta < \frac { \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$
A) $\frac { 2 } { 5 }$

B) $\frac { 4 } { 5 }$

C) $\frac { 1 } { 5 }$

D) $\frac { 3 } { 5 }$

Quizplus · Accepted Answer

To find $\sin(2\theta)$, we use the double angle formula: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. Given $\cos(\theta) = \frac{\sqrt{5}}{5}$, we first find $\sin(\theta)$ using the Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$. Substituting the given value of $\cos(\theta)$, we get $\sin^2(\theta) = 1 - \left(\frac{\sqrt{5}}{5}\right)^2 = 1 - \frac{5}{25} = \frac{20}{25} = \frac{4}{5}$. Since $0 < \theta < \frac{\pi}{2}$, $\sin(\theta) = \sqrt{\frac{4}{5}} = \frac{2\sqrt{5}}{5}$. Therefore, $\sin(2\theta) = 2\left(\frac{2\sqrt{5}}{5}\right)\left(\frac{\sqrt{5}}{5}\right) = \frac{4\cdot 5}{25} = \frac{4}{5}$.

Use the Information Given About the Angle θ,0≤θ≤2π\theta , 0 \leq \theta \leq 2 \piθ,0≤θ≤2π

Use the Information Given About the Angle $\theta , 0 \leq \theta \leq 2 \pi$