Using sine, cosine, and tangent on the New SAT
If you’ve read the first part of the series and watched the video, you might be asking yourself the following: Who cares what sine, cosine, and tangent stand for? Why is this even interesting? As for SOHCAHTOA, I couldn’t care less!
Imagine, that you know only one side of the triangle, say AB = 4. If you know the value of Θ, say 20 degrees, you can figure out any of the three sides. This is similar to what happens when we have a 45:45:90 triangle or a 30:60:90 triangle. But that limits us to just one kind of right triangle. By using trigonometry, we can figure out all three legs of a triangle, provided we know one other angle measure—besides the right triangle.
Now back to SOHCAHTOA. What the heck does that have to do with anything? Look above at the definitions of sine, cosine, and tangent. You’ll notice that I’ve bolded the first letter of each, as well as the sides relative to Θ, giving us:
[Sine Opposite Hypotenuse]
[Cosine Adjacent Hypotenuse]
[Tangent Opposite Adjacent]
So if you ever forget exactly what sine, cosine and tangent mean, in terms of the relationship of the sides (which is likely to happen!), just remember SOHCAHTOA.
Now let’s do an actual question:
(figure not drawn according to scale)
AC = 10 and Angle CBA is 25 degrees. What is the value of the hypotenuse to the nearest thousandth?
We are given the side opposite angle CBA. And we are looking for the hypotenuse. Therefore, we want to set up an equation, in which we divide opposite by hypotenuse:
opposite / hypotenuse
This relationship (remember SOHCAHTOA) gives us:
Sine 25 degrees = 10/x, where x is the hypotenuse.
Now is time to whip out those calculators! (Don’t worry: trigonometry will not appear on the no-calculator section of the New SAT. If they happen to, they will be more about conceptual understanding than having to compute impossible sums).
Sine 25 = .42 = 10/x
0.42x = 10
x = 23.67