What are the unknown angles in the diagram below?

Question

So I used the Pythagorean theorem to find the missing leg first.

$$a^2 + b^2 = c^2$$

$$a^2 + 6^2 = 10^2$$

$$a^2 + 36 = 100$$

$$100 - 36 = 64$$

$$\sqrt{64} = 8$$

Now I'm just lost from here. It was just recently that I started studying tan, cos, sin etc., so any help is appreciated.

Answer 1

Let $\theta$ be the angle at $A$. From the perspective of $A$, we see that the side $BC$ is the "opposite" side, and $AB$ is the hypotenuse, so the ratio that is relevant here is sine:

$$\sin \theta = \frac{\textrm{opposite}}{\textrm{hypotenuse}} = \frac{6}{10} = 0.6.$$

So $\theta = \sin^{-1} 0.6$, i.e., the inverse sine of $0.6$. Notice that we didn't actually need to calculate $AC$ for this (although your work showing that $AC=8$ is correct).

Answer 2

You got $AC$ correctly as $8$. Now one observation you should make is that, you can find the area of the triangle now! It would be $$\Delta=\frac12.8.6=24$$ So now you can use that $$\Delta=\frac12 ab\sin\theta$$ where $\theta$ is the angle between $a$ and $b$. So by using it, $$48=60\sin B$$ $$48=80\sin A$$ you can solve for $A$ and $B$ angles.