Solving for Angle-side-side triangle in simplest radical form

Question

Assuming Capital letters are angles, opposite legs are lower case

In triangle $XYZ$, $\cos X = 1/2,$ $\cos Y = -1/4$ and $x =6$. Find $y$ in simplest radical form.

So far I have been using $\cos$ (adj/hyp) to create a possible ratio, but as you see am having trouble with the negative. Any pointers?

Answer 1

Here are some hints/general facts that will help you solve for $\sin Y$ exactly and without using a calculator:

• Pythagorean Theorem: $\cos^2 \theta + \sin^2 \theta = 1$.
• For $0^\circ < \theta < 180^\circ$, $0< \sin \theta <1 $.
• For $0^\circ < \theta < 180^\circ$, $\cos \theta <0 \;$ (i.e. $\cos \theta$ is negative) if and only if $\; 90^\circ < \theta < 180^\circ$.

Once you know $\sin Y$ you can use the information about whether $Y$ is acute or obtuse and the $\sin^{-1}$ function to say what the measure of angle $Y$ is exactly.

Beware, there may be more information than you need/distractors in the problem as it is described.

Answer 2

I have entered my work here, this I believe should be the right answer. Confirmation please

Also is this the right way to format? $\cos^{1/2}$ ?

$X = \cos^{-1}(1/2)$, $Y = \cos^{-1}(-1/4)$

$X=60$, $Y= 104.4$ ($q1,q2$)

$\frac{\sin60}{60} = \sin104.4/y$, $y = 6\sin104.4/\sin60$