Evaluate without using a calculator.
-$\sin \theta$, if $\cos \theta = \frac { 4 } { 7 }$ and $\tan \theta

Question

Evaluate without using a calculator.
-$\sin \theta$, if $\cos \theta = \frac { 4 } { 7 }$ and $\tan \theta < 0$
A) $- \frac { \sqrt { 33 } } { 4 }$
B) $- \frac { \sqrt { 33 } } { 7 }$
C) $- \sqrt { 33 }$
D) $- \frac { 7 } { 4 }$

Quizplus · Accepted Answer

Given $\cos \theta = \frac{4}{7}$, we can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find $\sin \theta$. Since $\cos^2 \theta = \left(\frac{4}{7}\right)^2 = \frac{16}{49}$, we have $\sin^2 \theta = 1 - \frac{16}{49} = \frac{33}{49}$, so $\sin \theta = \pm \sqrt{\frac{33}{49}} = \pm \frac{\sqrt{33}}{7}$. The condition $\tan \theta < 0$ implies that $\sin \theta$ and $\cos \theta$ have opposite signs. Given $\cos \theta > 0$, $\sin \theta$ must be negative, so $\sin \theta = -\frac{\sqrt{33}}{7}$.

Evaluate Without Using a Calculator sin⁡θ\sin \thetasinθ , If cos⁡θ=47\cos \theta = \frac { 4 } { 7 }cosθ=74​

Evaluate Without Using a Calculator $\sin \theta$ , If $\cos \theta = \frac { 4 } { 7 }$