Tip #34 to Get a Top ACT Math Score (page 2)
If you were in ancient Greece, the word "trigonometry" would not be intimidating. It would mean "triangle measurement," and when your pal Plato came up and said, "Hey dude, let's do our trigonometry homework," you'd just hear, "Hey dude, let's measure some triangles." But since you don't live in ancient Greece, it sounds like he's saying "Hey dude, let's do our %*#$!^ homework."
But trig is just triangle measurement. It's the way that sides and angles in a triangle are related to each other. They gave these relationships names that made perfect sense in ancient Greece, but now they freak some kids out, so stay relaxed when you see the names. Here they are, and for half the ACT trig questions, they are all you need to know:
This means, for example, that the sin of an angle equals the ratio of the side opposite the angle, divided by the hypotenuse. There's a great way to remember these ratios: SohCahToa. It stands for Sin = Opposite over Hypotenuse, Cos = Adjacent over Hypotenuse, and Tan = Opposite over Adjacent. I recommend taking a break right now and marching around the room chanting "SohCahToa." You're sure never to forget it. SohCahToa is all you need for half of the trig questions on the ACT.
Remember from Skill 24 that "hypotenuse" is just a fancy word for the longest side of a right triangle; it's always the one opposite the 90° angle. The other two sides, the shorter ones, are called legs. And "opposite" or "adjacent" refers to the leg opposite or adjacent to the angle being used.
Let's look at this question:
Solution: Easy! SohCahToa! The question asks for tan Z. Tan = . That means opposite leg over adjacent leg. So tan Z = = 0.75.
- Standing on Turtle's back, Vince casts a 4-meter shadow, as shown below. The angle of elevation from the tip of the shadow to the top of Vince's head is 40°. To the nearest tenth of a meter, how high is the top of Vince's head?
- If sin A = , then which of the following could be tan A ?
- For right triangle ΔABC with dimensions in feet as shown below, what is sin B ?
- For right triangle ΔXYZ shown below, what is cos Z ?
- B We always want to mark all the info from the question into the diagram. And sometimes the ACT does that for us. Here all info is marked. Thanks, ACT. So don't get caught up in the complex wording of the question. Look at the diagram. It's a straightforward SohCahToa question. The key to SohCahToa questions is simply to determine which ratio we need for the question. All we have to do is look at the info given. If we are given opposite and hypotenuse, we use sin. If we are given adjacent and hypotenuse, we use cos. And if we are given opposite and adjacent, we use tan. That's it. So based on the 40° angle, we have the adjacent leg and want the opposite leg. So, we use tan:
- G Remember that sin, cos, and tan are talking about right triangles, so draw a diagram. Sin represents we know that the opposite leg of vertex A equals 3 and the hypotenuse equals 5. Whenever you have two sides of a right triangle, use a2 + b2 = c2 to get the third side. Great mighty Pythagoras review (Skill 24). So 32 + b2 = 52 and b = 4. Therefore, since the opposite leg equals 3, and the adjacent leg equals 4.
- A In the diagram, the leg opposite vertex B equals 5, and the hypotenuse equals 7. So
- J In the diagram, the leg adjacent to vertex Z equals 4, and the hypotenuse equals 5. So
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