Solving for Angle-Side-Side Triangle in Simplest Radical Form
$\Begingroup$ 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...
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?
2 Answers
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.
I have entered my work here, this I believe should be the right answer. Confirmation please
Also is this the right way to format? $\cos^{1/2}$ ?
$X = \cos^{-1}(1/2)$, $Y = \cos^{-1}(-1/4)$
$X=60$, $Y= 104.4$ ($q1,q2$)
$\frac{\sin60}{60} = \sin104.4/y$, $y = 6\sin104.4/\sin60$
Sarah Jenkins is a veteran tech journalist with over 12 years of experience covering artificial intelligence, mobile innovations, and digital ethics. Her insights have appeared in leading technology publications worldwide.
