ACT Crash Course
With Shmoop's ACT crash course, your pencils may be #2, but you'll be #1.
If what you need is an outlined plan to cram every Test Prep strategy imaginable in the form of a short course that you can do with your mom or your teacher, this is what you want. This ACT Crash Course will take you by the hand and teach you exactly what you need to do and exactly when you need to do it in order to ace the ACT. In fact, taking this ACT Prep Course will make you so good at ACT strategery that you could write the test yourself.
Here’s what else you’ll get with your low-cost ACT Crash Course:
- an outlined plan to take you through the 5 sections of the ACT exam: English, Math, Reading, Science, and Writing
- killer test-taking strategies to ace each section
- hundreds and hundreds of practice questions, each with a video explaining the question
- videos reviewing major concepts and sample problems
- two complete, timed practice exams
- a virtual classroom interface where grades and work are organized, teachers can virtually communicate with students, and the class can discuss material through the discussion board
If you’re an instructor looking for ways to put together a course for students (or just want to prove you can still get a perfect score of '36'), this course is also fully loaded with a virtual classroom interface to fit whatever situation you’re in.
*ACT is a federally registered trademark of ACT, Inc. Shmoop University is not affiliated with or endorsed by ACT, Inc.
Unit 1. The ACT English Test
A review of the major concepts on the ACT English test + 20 drill sets = the ticket to ACT English mastery.
Unit 2. The ACT Mathematics Test
Complete coverage of ACT Math, from Pre-Algebra to Trigonometry, with over a hundred videos to make learning a breeze.
Unit 3. The ACT Reading Test
From the Prose Fiction passage to the Natural Science passage, we've got the Reading section covered.
Unit 4. The ACT Science Reasoning Test
We review the major skills and question types appearing in the coolest part (Celsius) of the ACT: the Science Reasoning test.
Unit 5. The ACT Writing Test (Optional)
The ACT Writing test may be optional, but we're fans. We go over the expectations, help you out(lining), and go through sample essays in this unit.
Sample Lesson - Introduction
Lesson 6: Trigonometry
You heard it here first: triangles and circles are coming back with a vengeance this season.
By season, of course, we mean at this point on the ACT, which is trig. Trigonometry is most likely the freshest of the math topics in your mind, considering that it's probably the course you have most recently taken. It could also possibly be a course you haven't gotten to yet, meaning that you should pay particular attention to all the math-y goodness that's about to go down.
This time around, we have to apply some new skills to our mélange of trigonometry wonders and learn some new names like sine, cosine, and tangent, as well as secant, cosecant, and cotangent, Also: coelacanth. (Just for good measure.)
After we're through dealing with prehistoric fish, we will also be dealing with radians and the unit circle. Remember SOHCAHTOA? (Hint: Sine = opposite/hypotenuse, Cosine= adjacent/hypotenuse, Tangent = opposite/adjacent). Yeah, we thought you might.
Grab a refreshing beverage and do some stretching, because this last ACT math review section will have us doing some mental gymnastics.
Sample Lesson - Reading
Reading 2.6a: SOHCAHTOA
Trig, as we affectionately refer to trigonometry, is the study of ratios and right angles. Angles come in all shapes, sizes and, more importantly, degrees. We already know that right triangles have right angles and are congruent.
Similar triangles have proportional sides. From a philosophical standpoint, think about how one of your shoes could fit inside your dad's shoe way back in the day. It was proportionately smaller. One might even say it was "similar." In the same way (a similar way?) similar triangle have sides with the same ratios.
If we're looking a triangle with a side of 5 feet that's similar to a right triangle with a side of 10 feet, the widths would follow a 1:2 ratio. Once we know we're dealing with a right angle, other rules fall into place. Consider, for example, SOHCAHTOA, which speaks to the relationship between all the sides of a right triangle.
If you go no further in trig, this is the one concept you must know. Since you're stuck with us for the long haul, though, we might as well go all the way back.
In the beginning, there was the unit circle. This seems like a good starting point.
Draw any radius out of the center of the circle to the edge. As a result, we can find out the angle between the radius and the positive x-axis.
We find ourselves with a right angle by dropping a line down from the point where the radius meets the edge of the circle.
The hypotenuse of this triangle is the radius. The opposite side is the leg that is opposite the angle of the right triangle and touches the center of the circle. The adjacent side is the leg or the right triangle that is adjacent to the angle whose vertex is the center of the unit circle.
Once we know this, we have one cozy family of functions. The input of these functions is the angle between the positive x-axis and the radius; the output of the function is a certain ratio coming from the right angle. Without further ado, let's meet the family of sine, cosine, and tangent.
Seriously: if you've managed to memorize the names of all the Bradys on the Brady Bunch, trig will be a breeze.
The sine of an angle equals the length of the side opposite or across from that angle over the length of the hypotenuse. This is SOH, or sin – opposite – hypotenuse.
The cosine of an angle equals the length of the side adjacent to that angle over the length of the hypotenuse. This is CAH or cos — adjacent —hypotenuse.
Finally, we have tangent, or TOA. The tangent of an angle is equal to the length of the opposite side over the adjacent side.
If we want to get a little more shmantzy, we can look at secants, cosecants, and cotangents. The odds of you encountering more than one question on this triumvirate is almost zero, but it could happen. Better to be overprepared than underdressed, we always say.
Sample Lesson - Reading
Reading 2.6b: Solving Triangles
Right triangles are special, and we're not just saying that because we think all triangles are special. (Although our parents never pulled their punches when they were discussing favorites….awkward.) If you already know that the base angle is 90 degrees, a bunch of other elements fall into place.
Whenever we have a right triangle, we can determine the trig values of the various non-right angles:
When we do trigonometry, we have the choice between describing angles in degrees or radians, which help you diagnose your way around a circle. Sounds serious. The equation to remember is that 360° = 2π radians. Still, some of the other values are so important that you should always, always know their trig values:
sin (0) = 0
cos (0) = 1
tan (0) = 0
sin (90) = 1
cos (90) = 0
tan (90) = undefined
Note that 90 degrees is the same as π / 2 radians.
Note that π radians is the same as 180 degrees.
Note that 3π / 2 radians is the same as 270 degrees.
Contemplate an equilateral triangle:
Well, that was refreshing. But now we're bored, so we create a right triangle by bisecting the top angle:
We call this a 30-60-90 triangle, because it has one angle of each of these degrees. We could also call it a wonky-tastic triangle, but that's less descriptive and probably not a real math term. Probably.
The hypotenuse has length 1, and the side on the base has length 1/2. Using the Pythagorean Theorem, we find that the other leg has length
Given all of this information, we know that:
An isosceles right triangle is a 45-45-90 triangle.
Using the Pythagorean Theorem, we see that the hypotenuse has length
Where is this useful? You can use values you know to figure out the lengths of sides within a triangle that you don't know. It's so convenient that way, like pre-sliced bagels.
Consider the following right triangle. We're given the length of one side and the size of one angle, but what we really want is to determine the lengths of the other two sides.
To find x, make a ratio of the side x to the side of length 5 by using a trig function. Since the side x is adjacent to our given angle, and the side of length 5 is opposite this angle, we should try to use a tangent or cotangent. One way to do this is to observe that:
We could either use the Pythagorean Theorem to determine the length of side y, or we could do something else. Let's do something else.
y = 5 × 1.54 = 7.7
Sample Lesson - Reading
Reading 2.6c: Trig Identities
You know that "You Are Here" sign on the map at the mall? We do, but that's because we spent several of our formative years as mallrats. Anyway, that's the basic idea of trig identities: If you know where you're standing in relation to the organic tea shop, it's easy to find the place that sells the hermit crabs.
Man, they don't make malls like they used to.
Each of these identities represents a different way of identifying the actual place on the map where you're located. Put another way, if x = 3 and you have an equation x + 3, you can pull the ol' switcheroo to get 6. That kid from The Parent Trap managed to get a twin sister and a full set of parents, but we can't all be that lucky.
There are a number of equivalent statements for trig. If you're going for a 36 on the ACT—and you totally are—you'll need to learn them all. Otherwise, the first one is the most important. We know that lists are so yesterday, but we thought you'd want just the facts instead of our clever little stories.
Look, we're learning restraint! Isn't that cute? Anyway, the list;
sin2(θ) + cos2(θ)= 1
1 + tan2(θ) = sec2(θ)
1 + cot2(θ) = csc2(θ)
sin(α + β) = sin(α)cos(β) + sin(β)cos(α)
sin(α – β) = sin (α) cos(β) – sin(β)cos(α)
cos(α + β) = cos(α) cos(β) – sin(α)sin(β)
cos(α – β) = cos(α) cos(β) + sin(α)sin(β)
sin (2θ) = 2 sin (θ) cos (θ)
cos (2θ) = cos2 (θ) – sin2 (θ)
Sample Lesson - Reading
Reading 2.6d: Trig Graphs
You should also be familiar with the Big Three of trig graphs and how they might change, as all things tend to do eventually. We're getting a little nostalgic here.
A graph's period refers to how long it takes to repeat itself and its amplitude is its maximum.
A sine graph has points (0, 0), (π, 0), (,1), and (3π/2,-1). It has an amplitude of 1 and periodicity of 2π.
A cosine graph has points (0,1), (π,-1), and (2π,1). It has amplitude of 1 and periodicity of 2π.
A tangent consists of vertical asymptotes. Remember it can also be described as which always seemed a little greedy to us, but whatever. Whenever cos(x) = 0, we have a discontinuity in the graph. It repeats itself π times, or has period π.
These graphs can all be written in the following form where a = amplitude.
y = asin(x)
y = acos(x)
y = atan(x)
If our graph changes such that the maximum gets higher, then the a changes.
We sometimes shorten or lengthen the wavelength in our graphs. This affects the period, which is calculated as (2π/b).
y = asin(bx)
y = acos(bx)
A trick that might be helpful in considering these graphs is that when the height goes up (lengthens), a gets bigger. When the height shrinks, a gets smaller.
In terms of periodicity, when the graph crunches horizontally, b gets bigger. When the graph stretches out horizontally, b gets smaller. Think about it like you're pulling on a string.
A final next step is to shift the curves right or left. This can be done through the formulas.
y = asin(bx + c) and y = acos(bx + c). If you're moving left, or adding, c is positive. If you're moving right, or subtracting, c is negative. These don't come with value judgments; we're just saying.
The shift is calculated as -c/b. We don't want to repeat ourselves—lies, we love repeating ourselves—but the phase shift is left if -c/b is negative and right if positive.
Sample Lesson - Reading
Reading 2.6e: Trig Equations
Just as we learned to solve equations involving a variable x, common parlance in the Land of Trigonometry is to describe things in terms of θ (theta). We'd rather have this kind of theta, but what can you do? The ACT is a cruel mistress.
Consider the following trig equation:
sin(θ) = cos(θ).
We can divide by cos(θ) because we can verify that any θ for which cos(θ) = 0 will not satisfy this equation (since both cosine and sine are never equal to 0 at the same time). Dividing both sides by cos(θ), we obtain the equation:
When is tan (θ) = 1? This happens when:
Be careful: That isn't the only time this happens. Because of the periodicity of the tangent, this equation is also true whenever
where n is any integer.
If you can use your calculator, then you can instead solve by using the inverse trig function. In the problem above, we could instead look at:
θ = tan-1 (1), which is or 45 degrees.
Plugging this into our calculator will also provide the correct solution. You were skeptical for a minute, weren't you, though?
Sample Lesson - Reading
Reading 2.6f: Modeling with Trig Functions
Picture a pendulum. The pendulum has "periodic" behavior. That is, it repeats the same arc of motion over and over again…on Earth. Actions or calculations that repeat themselves are perfectly suited for using functions to describe their behavior. To write an equation, we need to have some formulas describing this type of back and forth dance.
In general, any time you have an action repeat itself, it is considered periodic, like the World Series, the Olympics, or the World Cup. It also makes it super exciting when you know they're coming back! Well, we think so, anyway.
Sound waves also exhibit periodic behavior. It turns out that any sound wave can be described as either a sine or a cosine. (Seriously. Even this one!) Mathematically, it can be written as:
s(t) = Asin(wt).
In fancy trig language, A is called the amplitude,and w is the frequency.
When modeling periodic behavior, always remember to ask how high and how far the period goes. Let's say, for example, that the Loch Ness monster as a baby had a tail that looped above the water, making loops that were only 3 feet wide and 2 feet high. As she grew into Nessie, the loops became 10 feet wide and 6 feet high. Notice that she got "flatter" over time.
This is because of the sad fact that as you age, no matter who you are, gravity always wins. True story.
You can also decide whether it's better to use a sine or a cosine based on if it starts at 0. In other situations (like…an ACT-shaped situation), you might be given a trig function that models some behavior. Suppose, for example, that the average temperature in a given month, measured in degrees Fahrenheit, is modeled by the equation:
where t is either 1, 2, .. , 12, representing the months of the year. Which month has the coldest average temperature?
To solve this question, we look at the trig function. We know that the sine function is its smallest when it is equal to -1, and this happens when it is evaluated at π/2. To find the month with the smallest temperature, we need to solve:
The coldest average temperature is in January. You might want to check and determine that the period for this function is 12 months…or that parka we packed for our summer vacation isn't going to be super useful.
There we have it. All the ins and outs of the ACT Math section. Even if some of this seemed like new information, there's no need to worry: review a few times, go through our handy drills, dabble with a few practice sections, and you'll be ready to tussle with ACT Math like a prizefighter. Get your game face on!