Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$, given that $\sin \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant II.
A) $- \frac { \sqrt { 7 } } { 9 }$
B) $- \frac { 3 \sqrt { 7 } } { 7 }$
C) $- \frac { 3 } { 2 }$
D) $\frac { 5 } { 4 }$

Question

Quizplus · Accepted Answer

In quadrant II, sine is positive, cosine is negative, and tangent is negative. Given $\sin \theta = \frac{3}{4}$, we can find $\cos \theta$ using the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$. Solving for $\cos \theta$, we get $\cos \theta = -\sqrt{1 - \sin^2 \theta} = -\sqrt{1 - \left(\frac{3}{4}\right)^2} = -\sqrt{1 - \frac{9}{16}} = -\sqrt{\frac{7}{16}} = -\frac{\sqrt{7}}{4}$. Then, $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{4}}{-\frac{\sqrt{7}}{4}} = -\frac{3}{\sqrt{7}} = -\frac{3\sqrt{7}}{7}$, after rationalizing the denominator.

Use the Fundamental Identities to Find the Value of the Trigonometric