Trigonometry Examples

Find the value of x.

Looking at the known angle, the cosine should take care of everything.

The hypotenuse is 10, the adjacent side is x, and ∠Q is 17°.

Rearrange to solve for x.

x = 10cos(17°)

x ≈ 9.6

What is the measurement of ∠D if tan D = 1?

If we're looking for an angle measurement, we know that we have to use inverse trig functions.

tan D = 1

tan^–1(1) = D

A calculator could easily give us the answer, but we can also think this one through just based on the ratio itself.

If we ignore the tan D part for a sec, something amazing happens.

opposite = adjacent

Hold up. That means both the legs of the triangle equal each other. In other words, we have a triangle that looks like this. Look familiar?

Look again.

That's right! It's a Gandalf (45-45-90) triangle, one of our magical triangles from the previous section. That means ∠D is a 45° angle. If we want, we can use the inverse trig functions and a calculator to solve it, but at least now we know why the answer is what it is.

Find the measure of ∠H in ∆GHI when the triangle has the following points: G (0, -5), H (-5, -5), I (0, 2).

As with all coordinates, the best thing to do first is to find the lengths between the points. We find drawing out the image quite helpful.

Using the distance formula () and/or looking at the grid can give us the lengths of the legs of the triangle.

GH = 5
GI = 7

If we use the tangent ratio, we don't have to worry about calculating the hypotenuse. Since we're looking for an angle again, it's best to use the inverse tangent.

The side opposite to ∠H is GI and the side adjacent to it is GH.

A calculator will give us our answer.

H ≈ 54.5°

What's the measure of ∠KMN?

We can't figure out the value of ∠KMN directly; those trig ratios only work for right triangles. 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 = 62sin(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 ∠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°