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ACT Trigonometry

Teacher/Instructor Devorah Goldblatt
Devorah Goldblatt

Case Western Univ., summa cum laude
Perfect scorer on the SAT & the ACT

Devorah is the founder of Advantage Point Test Prep and the author of the book “Boost Your Score” The Unofficial Guide to the Real ACT.

So hopefully I'm not outdating myself too much. If I tell you that when I was in High School the Makarena was really really big. You probably have heard it, you know with that dance that goes like, right, okay. But here's a question who's the artist who wrote the song? Everyone knows it, everyone knows the dance, who's the artist? You probably have no idea, let me tell you why, they never made another song that was a hit. So that's called a one hit wonder and in this episode we're going to talk about one hit wonders on the Math section. They're always going to be some questions that without fail will show up one time. And they look a little complicated unless you know exactly how to do them. So it's kind of like... if it's easy to learn why not learn and you can just nail those questions which you know will appear. In this episode, we're going to take a look at four one hit wonders and then afterwards we'll take a look at SOHCAHTOA which is a trigonometry question that can help you easily answer two out of the four trigonometry questions on the ACT.
Four one hit one hit wonders. You're always going to see a question about the fundamental counting principal, a question about matrices, a question about the equation of a circle and a question testing if you can find the area of a parallelogram. And you may be thinking oh my God I don't know how to do any of these things it just sounds really complicated. The reason I picked them out is 'cause they're actually very simple to learn. So we're going to quickly go through these and you'll see how you can really easily pick up these points on testing.
First, fundamental counting principle. Here I am in my closet deciding what to wear. I have 3 skirts, 5 shirts, 6 pairs of shoes and 2 pairs of socks. If I mix and match these elements, how many total different outfits can I make? An interesting question you've probably seen under the practice test. So okay students do all sorts of things you know, making a list of how many actions they have, no need. The fundamental counting principal says you just multiply all your different options and that's the total amount of combinations that you have. So here we go, it's just 3 times 5 times 6 times 2. Okay well 5 times 6 that's 30 times 2, 60 times 3, 180. So 3 times 5 times 6 times 2, 180 and we have 180 different kinds of outfits I can make with these combinations.
Let's look at at a more difficult problem. Once in a while you'll see these and these require an extra step, but you can still do it with no problem. How many different combinations are possible for a seven digit telephone number? Also just an interesting question, I'm kind of curious myself. Well, okay, think about what we did before, we thought about how many choices we had for each option and we multiplied that. Right we have... I'll take 3 skirts, 2 shirts, so how many choices for each and then multiply them. Here we know we have seven different options right? So seven different numbers in our telephone number, one, two, three, four, five, six, seven. And the question is how many choices for each one of those slots for the number? Okay, this is the part where it's harder; you have to think how many choices are available there? Well for each digit of a phone number, ten choices right, you've got zero, one, two, three, four, five, six, seven, eight, nine, that's ten. So how many different combinations are possible for seven digit number? Well ten options for everyone in the slots, so you would have 10 times 10 times 10 etcetera for each of the seven slots, so really 10 to the seventh power. Okay and we if we do that on a calculator, what would that look like? Well, 10 to the seventh is 10,000,000, okay that's answer choice C, perfect. So 10,000,000 different possibilities for a seven digit telephone number. Again we've found the amount of slots that we need to fill right, seven slots for the seven digits and then how many options for each digit, 10 and then we just did 10 times 10 times 10 times 10 times 10 times 10 times 10. That's the amount of total combinations that we have, great.
Next one hit wonder, matrices. Students tend to get really intimidated with these. They look at them maybe it's been a while since you've seen it, maybe you haven't seen this at all. Actually, they're really easy once you know what to do. You've got this funky looking shapes here with some numbers in it. All you need to do is combine them in whatever way they tell you and you're going to combine each number with its corresponding number in the other square. So here, what is A plus B? Okay, so here you want to add A and B and all you do is add each number to the corresponding number. So for example, 2 plus 0 is 2, 3 plus 2 is 5, and you know what, let's first look at the answer choices too and just make sure do we even have to keep going. 2 plus 0 is 2 and actually only one of these even has a 2 here. So 2 plus 0 is 2 but let's just double check. We said 3 plus 2 is 5, that looks great, 0 plus 1, 1 and negative 1 plus 1 is 0. So this is the solution for this matrix problem. So you see, not that complicated at all, nothing to be intimidated about. And once in a while you'll have subtraction and then you would just subtract each relevant part. So that's a matrix problem.
On to equation of a circle. So here we go, we've got our equation for our circle and this kind of a complicated circle thing to learn, but once you know what to do, you'll be able to get this question and it definitely will show up. x minus h square plus y minus k squared is r squared. In this equation all you need to know, h and k are the center of the circle, h is going to be the x coordinate of the center point and k is going to be the y coordinate of the center point, r squared that's the radius squared that's the equation of the circle. So you'll see a problem that looks something like this; a circle with a radius of 5 is placed on the coordinate plane so that it is centered at the point 1,2, this is our essential point, this is the h and the k that we care about. What's the equation of the circle? Okay, so all you would do is just plug these in to the equation for the circle. We know the center is h and k, h is the x and k is the y coordinate, so here we're going to plug in our 1 here, x minus 1 squared, we're going to plug in out 2 here, y minus 2 squared and our radius is 5 so remember 5 squared which would be 25. Let's look at some answer choices, okay again we want x minus 1 squared right, then we want y minus 2. If you look at the answer choices, keep an eye out for this, they know there going to be students who forget the radius has to be squared. So there's always going to be some answer choices with pure un-squared radius. And here we go we already know C and D are out right, they're just 5. Also by the way, when we talked about strategies we talked about getting rid of the misfits and by the way, E is negative and it's the only negative one of all the answer choices, so that can't be right either. Okay, if you look at A and B which one of these fits the equation? Here we go, x minus 1 and y minus 2 right? It was x minus h, y minus k and we plugged in our 1 and our 2 for h and k and here we go we have a radius squared which gives us 5 squared 25. So this is the equation of our circle.
Next one hit wonder, area of a parallelogram. So this is a common parallelogram area question you would see. What's the area parallelogram ABCD and you've got a parallelogram with some of the side lines in it. Students do some really funky things with these, they just forget, it's been a while since you learned the area of a parallelogram. I've seen students divide it up, they make it into a square with two triangles all sorts of things not necessary. It is so easy to find the area of a parallelogram. It's just base times height, so just base times height. Okay we talked about a height and a base, we talked about how a height hits the base at 90 degrees. So here we've got our base which is 6, we've got our height, which is hitting at a 90 degree angle okay. So we know we need to find this length, we also talked about triangles when we talked about geometry so we have a right triangle here, you've got two of the sides you can find the third using the pythagorean theorem. We know that 3 squared plus this side squared is going to give us 5 squared. Okay let's write this out, so 3 squared or 9 plus, that side that we're missing, we'll call it x squared, is equal to 5 square right so 25 okay. In that case x squared is equal to 25 minus 9. So x squared is equal to 16, so if x squared is equal to 16, x is just equal to 4, x is equal to 4. I'll just write that here and we can write that in and here we go. We've got our height, we've got our base and we can find the area of our parallelogram, just 4 times 6, 24. So B is correct here.
And that's it for one hit wonders and you see now, they are not as intimidating as they look. If you go over this concepts a little bit more, you'll be able to get these questions which you'll know will show up on the ACT.
Let's move on and talk about SOHCAHTOA. So there are four trigonometry questions on the ACT and two of them are going are to be simple, two of them are going to be pretty complex. The cool thing is that if you know SOHCAHTOA which we'll talk about in a second, you can easily answer two out of the four questions. And now if you're interest in how to answer the other two questions, we do have a good tutorial for that in the bonus material. But right now we're going to talk about how to easily use SOHCAHTOA to find those two easier trig questions. So let's review SOHCAHTOA. SOHCAHTOA stands for Sine is the opposite over the hypotenuse, Cosine is the adjacent over the hypotenuse and Tangent is the opposite over the adjacent. And now if you want it written out it is right here. But it's good to remember just that whole acronym, how it's spelt and what the different parts stand for. Okay so we're going to want to find the Sine, the Cosine and the Tangent of an angle and the question is how do you know? Students always ask me, how do I know what's opposite? What's adjacent you know for a particular angle? So let me just show you, lets say we care about angle A here and they'll always tell you what angle you need. So you need to know what's the adjacent, what's the opposite, what's the hypotenuse. Well the hypotenuse is always the side opposite the right angle no matter what. So just to review, the opposite side is the side opposite the angle, you'll feel it, it's pretty opposite, it's far on the other side. And adjacent is always going to be connected to the angle you care about. You know you've heard the phrase, let's say the adjacent building, it's connected so that's how you'll know this is your adjacent this is your opposite. Let's take a look at a sample question. What's the cosine of angle B? So keep in mind angle B right here, we'll mark it, is the one we care about. Cosine, where does that come in in SOHCAHTOA? SOH CAH, the middle. C A H right? SOH CAH TOA. Okay, so cosine is adjacent over the hypotenuse, that's what that stands for. So adjacent over hypotenuse okay, in relation to angle B, adjacent remember attached, so 4 over the hypotenuse which would be 5 so your answer would be 4 over 5 answer choice C, great! There's a lot more practice with SOHCAHTOA in your bonus material so if you feel like you need a review you may want to head there next.
Let's recap. We talked about one hit wonders and how you know they're going to show up, they're not as intimidating as they look. So it's good to just have the skills to be able to tackle them on testing. And we talked about SOHCAHTOA, the one trigonometry concept that can help you easily tackle two out of the four trig concepts you'll see on the ACT.