We can't figure out the value of ∠*KMN* directly; instead, we'll have to be sneaky about it. Since ∠*KMN* is part of a larger angle, subtracting ∠*LMK* from ∠*LMN* should give us the right value. m∠*KMN* = m∠*LMN* – m∠*LMK* Focusing on the first angle, LMN, we know the hypotenuse and the adjacent side. Since it's the angle we're looking for, that translates to inverse cosine. Add in what we know. Plugging it into our calculator gives us: ∠*LMN* ≈ 49.8° We're off to a great start. The second angle is going to be a bit trickier. Before we can calculate m∠*LMK*, we'll need to know something else about the smaller right triangle. Since the unknown leg *LK* is half of the larger right triangle's leg LN, we can start by finding *LN*. The Pythagorean theorem would work just as well, but we'll stick to trig.
Rearrange to solve for *LN*. *LN* = 62 × sin(49.8°)
*LN* ≈ 47.4
Great, but we were actually looking for *LK*. Dividing *LN* by 2 will give us *LK*.
*LK* = 23.7
Wonderful. The whole point of calculating *LK* was so we could find the ∠*LMK*. Now with both the legs of ∆*LKM* found, we can use the inverse tangent to find ∠*LMK*. Remember, we're looking at the smaller triangle with respect to ∠*LMK*. That means our opposite side is *LK* and our adjacent side is *ML*.
Substitute in those values and solve.
∠*LMK* ≈ 30.7°
Finding ∠*KMN* hasn't been an easy task, but now that we know ∠*LMN* and ∠*LMK*, we can do it. Finally. m∠*KMN* = m∠*LMN* – m∠*LMK* Plug in our values. ∠KMN = 49.8° – 30.7° ∠KMN = 19.1° |