Explanation:
In my explanations, I'll refer to the three sides as BC, AC, and BA. BC is the same as saying side A, AC is the same as saying side A, and BA is the same as saying side C.
As you've correctly discovered, you can use trigonometry to find the measures of angles a and b.
Angle A:
- When angle A is the reference angle, side BC is the opposite side and side AC is the adjacent side.
Thus, we have tan (θ) = opposite / adjacent.
When we substitute 52 for the opposite side and 48 for the adjacent side, we have tan (θ) = 52/48, where
- θ is the measure of our reference angle, namely angle A.
- As you've seen, we must use arctan to find the measures of angles:
arctan (52/48) = θ
47.2906100426 = θ
47.3 = θ
You rounded to the nearest tenth and this is how you found that angle A = 47.3°.
Angle B:
- When angle B is the reference angle, side AC is the opposite side and side BC is the adjacent side.
Thus, we again can use tan (θ) = opposite / adjacent.
When we now substitute 48 for the opposite side and 52 for the adjacent side, we have tan (θ) = 48 / 52
To find θ (the measure of angle B), we must use arctan:
arctan (48 / 52) = θ
42.7093899573
You also rounded to the nearest tenth for this and that is how you found that angle B = 42.7°.
Side BA (the hypotenuse):
- Because this is a right triangle, you remembered that we're able to use the Pythagorean theorem to find the length of side BA (the hypotenuse).
The Pythagorean Theorem is given by
a^2 + b^2 = c^2, where
- a and b are the shortest sides called legs,
- and c is the longest side called the hypotenuse.
Thus, as you've written, we can find c by plugging in 52 for and 48 or b in the Pythagorean Theorem. Then, we'll take the square root of the sum of squares of 52 and 48 to find c, aka side BA (the hypotenuse):
52^2 + 48^2 = c^2
2704 + 2304 = c^2
5008 = c^2
√5008 = c
70.7672240518 = c
70.8 = c
Thus, you rounded to the nearest tenth and this is how found that side BA (aka side C) is 70.8 units long.
I would put units instead of ° for you answer since units are for side lengths and ° are for angles.