How do I use SOHCAHTOA in trigonometry?

I've heard about the SOHCAHTOA acronym for trigonometry before, but I'm not really sure when to use it. Can anyone give me a brief explanation on how and when it's applied in solving trig problems?

SOHCAHTOA is a mnemonic device used to help remember trigonometric ratios in right-angled triangles. It stands for Sine (opposite / hypotenuse), Cosine (adjacent / hypotenuse), and Tangent (opposite / adjacent), which are the basic trigonometric functions. Here's a breakdown of the acronym:

- S = sine

- O = opposite side of the angle

- H = hypotenuse (the longest side of the triangle, opposite the right angle)

- C = cosine

- A = adjacent side of the angle

- T = tangent

- O = opposite side of the angle (again)

- A = adjacent side of the angle (again)

You can use SOHCAHTOA when you're given an angle plus one side length of a right-angled triangle, and you need to find an additional side. It's important to remember that this only works for right-angled triangles.

Example: Let's say you're given a right-angled triangle where one of the acute angles (not the right angle) is 30°, and the adjacent side to the angle measures 10 units. Since you want to find the length of the opposite side, you would use the tangent function (TOA):

tan(30°) = opposite / adjacent

tan(30°) = opposite / 10

Now, plug in the value of tan(30°), which is 1/√3 or (√3)/3:

(√3)/3 = opposite / 10

To find the length of the opposite side, multiply both sides by 10:

opposite = 10 [(√3)/3] = (10√3)/3

So, the length of the opposite side is (10√3)/3 units.

In sum, SOHCAHTOA is a helpful tool for solving right-angled triangle trig problems when given an angle and one side length. Make sure to identify the right trig function to use according to the given information, and apply the mnemonic to easily solve the problem.