Trigonometry Question, Please Help

Question

Find $\sin\theta$ if $\tan\theta$=4; 180< $\theta$ <270. I just don't know what to do. I'm not sure if I missed a section or what. I have online Trig and it's confusing.

Answer 1

$$180^\circ<x<270^\circ\implies \sin\theta<0$$

$$\sec^2\theta=1+\tan^2\theta$$

$$\sin^2\theta=\frac{\tan^2\theta}{\sec^2\theta}$$

Answer 2

Think geometrically.

$\sin \theta$ is the $y$ coordinate of some point $P_\theta$ on the unit circle. Since $180^\circ < \theta < 270^\circ$, we know that $P_\theta$ is somewhere to the southwest of the origin; in particular the sine we're looking for is negative.

You also know that $P_\theta$ is on the line that passes through $(0,0)$ and $(1,4)$, since $\tan\theta=4$.

Therefore the triangle that consist of $(0,0)$, $P_\theta$ and the point on the $y$ axis directly outside $P_\theta$ (with coordinates $(\sin\theta,0)$) is similar to the right triangle with corners $(0,0)$, $(1,4)$, $(0,4)$. The large right triangle has side lengths, $1$, $4$, $\sqrt{17}$, and if we scale it down by $\frac1{\sqrt{17}}$ so the hypotenuse becomes $1$, the longer of its legs becomes $\frac{4}{\sqrt{17}}$.

Therefore the sought sine is $-\frac{4}{\sqrt{17}}$.

Answer 3

I made a big mistake in my previous solution, so I will give a more proper hint now!

$\bf{HINT}:$ We know that our $\theta$ is in the third quadrant because of the position of the angle, and I am guessing you learned about the CAST system that indicates the sign of the trigonometric ratios, (e.g., the first quadrant is positive for any trigonometric ratio), so in the third quadrant the tangent of an angle is positive, while both $\sin$ and $\cos$ will be negative. So remember what tangent is, $\tan\theta = \frac{opposite}{adjacent}$ which means that $\tan\theta = \frac{opposite}{adjacent} = 4$ so the easiest way to optain this value is $\frac{-4}{-1} = 4$ (the negatives come from the fact that in the third quadrant we have negative $x$ and $y$ values). From there it shouldn't be too hard to find $\sin$.