multiplication

do you remember having to learn how to foil? you multiply the terms of a binomial or complex number in this order: first, outer, inner, last. let’s take a look at how to do it with a complex number.

that leaves us with this:

now remember, , as we already covered. so we get this:

division

all right, here’s where things get a little tricky, but stick with me. i promise, we’ll come out on the other side (mostly) unscathed.

let’s say you had to divide 5 + 2i by 6 + 3i.

now, remember, i is just another way of writing √-1. and, according to the ancient laws of math, we can’t have a radical in the denominator (or bottom part) of a fraction. so, it looks like we have to simplify in order to solve this problem.

step one: conjugate

in order to divide complex numbers, what you have to do is multiply by the complex conjugate of the denominator. i heard about half of you get sudden migraines there, but i promise, that’s not as complicated as it sounds. the complex conjugate is just the same exact denominator with one tiny change. instead of 6 + 3i, we take 6 – 3i.

so our problem now looks like this:

really, all we’re doing is multiplying by a fancy form of 1, so we’re not actually changing the problem; we’re just simplifying it.

step two: multiply

it looks like we’re out of plastic wrap, which is okay, because all we need is foil. yes, the good old first-outer-inner-last method of multiplying binomials and complex numbers is back again. and this time, it’s personal.

okay, not really. but let’s foil anyway. we’ll do the numerator first.

that leaves us with this:

and now, do the denominator the same way:

step three: simplify

here’s our problem so far:

we already know that , so let’s change that in both the numerator and the denominator.

and now, combine like terms! watch the magic!

notice how the denominator suddenly doesn’t have any more i in it. we’ve fully simplified this problem! woo-hoo! take a nice deep breath, magooshers! you’ve earned it.

the post complex numbers on the act: multiplication and division appeared first on magoosh blog | high school.

okay, i know most of you aren’t reacting that way (…yet), but i promise you’ll quake in fear just a little less when this is all over.

trig is definitely the most intimidating math to most act students – that’s because most of them haven’t seen it before (or if they have, usually only in a cursory way). the good news: if you can memorize 1 acronym, 2 formulas, and 1 definition, you’ll be all set to tackle even the hardest problems!

table of contents

1. first of all: what is trigonometry?
2. the basics of trig for act math
3. practice with sohcahtoa on act math
4. reciprocal trig identities
5. some helpful hints for act math trig
6. practice makes perfect
7. graphing sine, cosine, and tangent
8. sine, cosine, and tangent graphs in practice: transformations
9. the unit circle
10. act unit circle practice

note: for more helpful math formulas, including ones for trigonometry, look here

first of all: what is trigonometry?

trigonometry is the field of math that deals with triangles–specifically, the relationships between the three sides and the three angles that make up every triangle. act trig is primarily concerned with right triangles. if you like right triangles, you’re going to do well here.

and typically the first thing you study in a trig class are right triangles:

so here’s a right triangle. let’s say that we are looking at the angle the arrow is pointing to. the side next to it is the adjacent side, the side opposite it is the opposite side, and the hypotenuse is, of course, the hypotenuse.

the basics of trig for act math

to help illustrate my next point, let me tell you a brief story.

once upon a time, there was a young man. he wanted to practice his baseball skills, so he started with throwing and catching. first he threw golf balls high into the air to see how high he could throw them without missing a catch. after a while, he got quite good at throwing golf balls, so he moved on to tennis balls. once he felt confident enough with tennis balls, he moved to actual baseballs. again, he became quite skilled at throwing baseballs, and decided to practice with bowling balls to keep improving his arm.

of course, throwing bowling balls straight up into the air is not, generally speaking, standard practice for an aspiring baseball player, and he dropped the bowling ball directly onto the big toe of his right foot. he went to the hospital and met a lovely german doctor who told him that, luckily, his toe wasn’t broken, but he would have to take care of himself until he healed completely. he asked the doctor what he should do to take care of his foot. the doctor replied, “you must sohcahtoa.”

i know, i know, that was terrible. i hang my head in shame for the awfulness of that joke. but seriously, sohcahtoa is the answer to your trigonometry fears. it is an acronym that tells you everything you need to know to figure out basic trigonometry problems. it means:

sine = opposite / hypotenuse (soh)

cosine = adjacent / hypotenuse (cah)

tangent = opposite / adjacent (toa)

so, if you were looking for the cosine of a particular angle, you would take the value of the adjacent side to the angle and divide it into the value of the hypotenuse. remember to keep things from the right point of view. opposite always means “opposite to the angle you’re being asked about” and adjacent always means “next to the angle you’re being asked about.”

practice with sohcahtoa for act math

example 1:

what is the sin of a?

knowing sohcahtoa, you would be able to answer that it is opposite/hypotenuse or . easy as that!
example 2:

what is the length of xz?

knowing sohcahtoa means that if we are given a right triangle with one known length and one known acute angle (meaning not the right angle) we can always find the other two lengths.
so in this case we can use sine to find the length of the hypotenuse.

sin(10) = 3 / xz
xz = 3 / sin (10)

we can divide sin of 10 degrees by 3 in our calculator to get the answer: approximately 17.28.
example 3

here’s one that’s just a teensy bit harder, but we are just going to apply the same principles.

the tree below casts a shadow that is 24 feet long, and the angle of elevation from the tip of the shadow to the top of the tree has a cosine of . what is the height of the tree?

the problem tells us that the cosine of the angle of elevation is . remember sohcahtoa, so we are concerned with the adjacent side over the hypotenuse. the fact that the cosine is means the ratio of the adjacent side to the hypotenuse is . so we can set up a proportion:

=

cross-multiplying to solve for x gives us x = 30.

but remember that this is the hypotenuse and we need to find the length of the vertical side to find the height of the tree. we can use the pythagorean theorem to find the length of the vertical side.

+ =

=18

so the height of the tree is 18 ft.

if you recognized that we had a 3-4-5 triangle in the beginning, you could actually take a shortcut and just use tangent of the angle of elevation to figure out the height.

reciprocal trig identities

you will definitely encounter questions that require you to use sohcahtoa, and you may encounter questions that ask about reciprocal trig identities. each of the three basic trig identities has a corresponding reciprocal trig identity:

cosecant = hypotenuse / opposite

secant = hypotenuse / adjacent

cotangent = adjacent / opposite

notice how sine and cosecant are the same except the numerator and denominator is flipped. that’s what we mean by reciprocal. it’s easy to remember that “tangent” and “cotangent” are reciprocals since they sound so much alike, but how what about the other two? i once had a math teacher who used, “co-co no go” as a mnemonic device to help my high school class remember. what he meant was that your brain may think that “cosine” and “cosecant” are reciprocals since they both begin with the prefix “co-“ but that isn’t true. “sine” goes with “cosecant” and “cosine” goes with “secant.”

some helpful hints for act math trig

let’s revisit our lovely little triangle:

you should know that you can do this:

this is what’s called the law of sines. usually, if you have to use this formula, the question will give it to you, but it’s a handy tool to have in your pocket.

next up is a nifty little equation that you can use on any angle. we’ll follow mathematical convention here and use the symbol θ (pronounced “theta”) to stand in for the value of the angle.

to translate from math back into english, the sine of any angle, squared, plus the cosine of any angle, squared, equals 1. could be useful if you’re trying to figure out a tough problem on test day, no? if you see this equation anywhere on your math test, just remember that it’s equal to 1.

and to round out our helpful hints, here’s one last equation for you:

translation: the tangent of any angle equals the sine of the angle divided by the cosine. so if a problem ever asks you to divide the sine by the cosine, you can just plug the tangent right in! (and you can figure out the value of the tangent by using sohcahtoa!) easy!

to recap, let’s look at the equations all in one place:

finally, an unusual definition to learn! a radian is another way of measuring an angle. some harder problems on the act will use radians instead of degrees.

there are 2π radians in one circle. each point on a circle corresponds to a certain number of radians.

this is used in trig to determine the location of the right triangle (and thus the negative or positive values of the sides). for example, if a trig question told us that angle theta is between 3π/2 and 2π, we know that the angle must be in the 4^th quadrant of the circle.to convert degrees to radians, simply multiply by π/180.

practice makes perfect

with all of this in mind, let’s do a sample problem!

the correct answer is… d! let’s walk through it.

to find the sine of ∠a, you need to know the values of the opposite side (line bc) and the hypotenuse (line ac). you know the hypotenuse is 8, but the problem didn’t give you a value for line bc. it did give you line ab, though, which is 6. so we can use the pythagorean theorem to figure out line bc!

is the pythagorean theorem, as you might recall from the review on triangles. substitute in the values we know, and it becomes:

now that we know the value of line bc, we can figure out the sine of ∠a.

and we have our answer!

graphing sine, cosine, and tangent

these graphs are usually graphed and expressed in degrees, but you may also see them expressed in radians.

sine and cosine both have standard graphs that you need to memorize for the act math test. the standard equation for sine looks like this: y = sin x. the “period” of the wave is how long it takes the curve to reach its beginning point again. the coefficient in front of “sin” (here 1), is called the amplitude. it effects how high and how low the wave reaches vertically. if that coefficient changes, then the height changes. for example, y = 5 sin x, would show a curve that reaches +5 on the y-axis and extends down to -5 on the y-axis.

cosine also has a standard equation. it looks like: y = cos x. for the graph of cosine, notice how it begins at its highest y-value and descends, whereas sine begins at the origin. cosine and sine have the same period of 2π. questions involving trig graphs will likely require you to match given equations with graphs, or interpret the meaning of certain graphs, such as in a question like this: what is the smallest positive value for x where y = cos 2x ?

the difference between y = cos x and y = cos 2x is that the coefficient in front of x is halving the period, so it will now take just one pi to complete its cycle. the smallest x-value for cosine usually occurs at π/2. for the new graph, it will occur at π/4, which is ½ of π/2.

confused? want more? kristin’s here to explain!

sine, cosine, and tangent graphs in practice: transformations

the act may also ask you to transform, or change, the equations and then explain what happens. using sines as our example, let’s take a quick look:

• the graph of y = 2 sin x increases the amplitude of the wave, meaning the waves are taller.
• the graph of y = ½ sin x decreases the amplitude of the wave, meaning the waves are shorter.
• the graph of y = sin 2x decreases the period of the wave, meaning the wave peaks are closer together horizontally.
• the graph of y = sin ½ x increases the period of the wave, meaning the wave peaks are farther apart.

what does that look like in practice? so glad you asked…

how will the graph of the function f(x) = 4sin x + 0.2 differ from the graph of f(x) = sin x?

a) the graph’s period will be 4 times as much and the graph will shift 0.2 units down.

b) the graph’s period will be 4 times as much and the graph will shift 0.2 units up.

c) the graph’s amplitude will be 4 times as much and the graph will shift 0.2 units down.

d) the graph’s amplitude will be 4 times as much and the graph will shift 0.2 units up.

we know that the coefficient in front of sine changes the amplitude, so (a) and (b) can quickly be eliminated, since 4 multiplies the amplitude by 4. just like a linear equation, adding to the end of an equation shifts a graph upwards. for example, the only difference between y = 8x and y = 8x + 7 is that the latter is 7 places higher on the y-axis. the answer is (d).

the unit circle

now, let’s take a look at the famous unit circle: a cool little circle that is often featured on act trig questions, so it’s a must-know act math thing.

this is a unit circle. it’s a circle with radius of 1 centered about the origin.

there are a lot of fascinating aspects to the unit circle: i suggest you consult the interwebs or your math teacher to find out more. we’re just going to go through the absolute basics here that will help you get some act trig questions right.

the act will test whether you know where angles larger than 360 degrees lie, and the unit circle helps us visualize this.

there are 360 degrees in a circle, but we can just keep swinging the arm of the angle around counterclockwise (just like the hands of the clock) to get to an angle bigger than 360. so, for example, if you want to know where an angle of 760 would be, you would circle around the circle twice (for a total of 720 degrees) and we would have 40 leftover degrees. so that angle would lie in the upper right quadrant of the unit circle (quadrant i).

the act will also often use radians on trig questions, and the unit circle helps us wrap our heads around this.

you should know that:

90 degrees on the circle = π/2
180 degrees = π
270 degrees = 3π/2
360 degrees =2π

the act will also test whether you know where the sine, cosine, and tangent of angles are positive or negative on the unit circle.

check out the video below for the actual math explanation of why sine, cosine, and tangent are positive or negative in certain quadrants!

there’s a great mnemonic to help you remember where trig functions are positive or negative:

all students take calculus

this helps you remember that:

in quadrant 1 → all (sine, tangent, cosine) are positive
in quadrant 2 → only sine is positive (and cos and tan are negative)
in quadrant 3 → only tangent is positive (and sin and cos are negative)
in quadrant 4 → only cosine is positive (and sin and tan are negative)

act unit circle practice

now let’s look at a test example to show you how this helps you out on a frequently-occurring act question:

if the value of cos x = -0.385, which of the following could be true about x?

a. 0 ≤ x < π

b. π/6 ≤ x < π/3

c. π/3 ≤ x < π/2

d. π/2 ≤ x < 2π/3

e. π/3 ≤ x ≤ 2π

using astc (all students take calculus), we can figure out where cosine is negative and narrow down our answer choices. cosine is negative in q2 and q3 and positive in q1 and q4. so we can eliminate any values that would fall in either q1 or q4. answer choices a, b and c all have values that fall in q1. answer choice e is q4. so that means the answer has to be d because that is the only answer choice in q2 where cosine is negative. (and yes, it often is as easy as that on the act!)

for more on unit circles, check out kristin’s video explanation here:

looking for more problems to polish your skills? check out these act math practice problems!

the post act math trigonometry appeared first on magoosh blog | high school.

• you can use a calculator on the act math test. you cannot use a calculator on the science test. (or the english and reading tests… not that you’d want to.)

• make sure the calculator you want to use will be allowed by the act proctors. you can use any four-function, scientific, or graphing calculator, unless it has prohibited algebra software on it. for full details on which calculators are not allowed, please look here.

• make sure to use a calculator you’re familiar with. don’t go out and buy a new calculator the night before the test. you need to know how to use it, and you need to be comfortable with all of the functions you might need.

• replace your batteries the night before the test, even if they’re not that old; the last thing you want is a dead calculator in the middle of your math section.

• please don’t use the calculator for checking extremely basic arithmetic (e.g., double-checking that 10 – 2 = 8). it wastes your time.

• please don’t use the calculator when the answer choices all have fractions or radicals in them. the calculator isn’t going to help you get to those answers anyway, so you’re wasting your time.

• pick up your calculator only when you’re sure of how to solve the problem and how the calculator will help you do so. using it any other way is a waste of time.

• remember that act problems are meant to be solved quickly. if you’re picking up your calculator to try to solve √597, you’ve probably made a mistake somewhere. double-check your work up to that point.

• just remember, you are the one taking the act. your calculator is not. the act math section is testing your math skills, not how well you can use a calculator.

now, there’s one final thing i want you to do. if you have the calculator you want to use for the act, i’d like you to enter this problem into your calculator:

1 + 2 x 3

if you get the answer of 7, congratulations! your calculator follows the standard order of operations. you probably learned this as pemdas, or “please excuse my dear aunt sally.” it’s the order that you’re supposed to follow when solving an equation: parentheses, exponents, multiplication, division, addition, and subtraction. all of the math problems you’ll be dealing with on the act will follow this order, and your calculator already takes that into consideration automatically. that makes things easier for you.

if you entered the problem into your calculator and it came back with 9, then your calculator does not follow the order of operations. this means a little more work for you; you will have to separate problems out into the proper order yourself before entering anything into your calculator. so, to get the right answer, you would have to enter the above problem into your calculator like this:

2 x 3 = 6

6 + 1 = 7

the post act tip: how to use your calculator wisely appeared first on magoosh blog | high school.

no, real-life imaginary numbers (and isn’t that a weird turn of phrase) were discovered/invented as a way to take the square root of a negative number. with real numbers, we can’t do that, but by using our imaginary number, we totally can!

basically, if you take the square root of -1, you wind up with the imaginary number i

translated into math, it looks like this:

simple so far, yeah? good, because we’re about to introduce a little complexity up in here.

simplifying imaginary numbers on act math

okay, let’s take a look at an example. say you needed to find the square root of -16.

well, how do you get the number -16? one way is to multiply 16 and -1, right? so let’s rewrite it like this:

we already know that √-1 is i, and you should know that √16 is 4. so now our example looks like this:

and there’s our answer!

simple, right? no? okay then, let’s try another example. what if we wanted to find √-x? using the same steps we followed above, here’s our progression:

a little more complicated now, but still totally doable. your answer would be i√x.

powers of imaginary numbers

all right, stop looking at me like that. this is way easier than it sounds at first.

let’s say you needed to figure out what is , or, to write it another way i x i. since we already know that i is √-1, asking for is just another way of saying . when you square a square root, both of them cancel out, leaving you with the answer of -1.

so, .

how about ? that would be . but instead of thinking of it as (√-1)*(√-1)*(√-1), think of it as building on the power we already know, namely . so would look like this:

which is just another way of saying

-1 x i

which is just another way of saying

-i

so, .

easy peasy so far, right? let’s try . we can break that down like so:

as we already know, is -1. this is just a fancy way of writing

which, of course, equals 1.

now let’s live dangerously and figure out . and remember, we’re building on what we already know, so we can rewrite it like this:

we already know that , so our example now looks like this:

or, to simplify:

i

a nifty little trick

let’s write out everything we’ve learned so far in a list, including

take a close look at everything we’ve found so far. the powers of i work in a very clear cycle. you can remember it with a lovely little mnemonic device that i learned in school. say the answers we uncovered aloud, ignoring the negative signs.

all you have to do is remember “i won,” and you can figure out any power of i, any time, anywhere.

but wait! i hear you cry. what about the negative signs? well, just remember that the negative signs are on the inside of the cycle, like so:

now that we’ve reviewed all that, let’s look at an act-style example.

the imaginary number i is defined such that . what does

equal?

1. i
2. -i
3. 0
4. -1
5. 1

i’ll give you a minute to try to puzzle this one out. if you need me, i’ll be humming the song they play on “jeopardy!”

.

.

.

okay, how did you do? did you remember “i -won, -i won”? let’s take a look.

the first four terms in our question work out like this:

i + -1 + -i + 1 = 0

you can count on the sum being 0 every four terms, since we’re working with a cycle. so let’s think about this logically. we’re dealing with at the end of our question. how many groups of four are there in 21?

as far as we’re concerned, that 0.25 left at the end is all we care about. that means we’re one term into the next cycle. and if you remember your mnemonic device, we’ve gone through “i -won, -i won” five times in this problem. as the first term is always i, our answer is a.

here’s one to try on your own:

the imaginary number i is defined such that . what does equal?

1. i
2. -i
3. 0
4. i-1
5. 1-i

remember our cycles of four. how many of those are we dealing with? 54/4 = 13.5

disregard the first 13 cycles and just look at the 0.5 at the end. that means we’re two terms into the 14th cycle, and everything before that cancels out. i^53 = i and i^54 = -1, so the remaining sum is i-1, so our answer is d. remember “i -won, -i won,” and it’s easy!

for more on imaginary numbers, check out kristin’s explanation below!
 

the post imaginary numbers on the act appeared first on magoosh blog | high school.

worried about act vocabulary? worry no more! we’ve collected the top 20 act vocabulary words you should know—along with their definitions and parts of speech—in one place for your studying ease. without further ado…

top 20 act vocabulary words*

*as i mentioned in the top tips post, the act will more commonly test secondary definitions of common words than esoteric or abstruse vocabulary. bear in mind that the second and third definitions of the words on this list are important, too! if you need more vocab practice, check out magoosh act prep.

the post top 20 act vocabulary words appeared first on magoosh blog | high school.

top tip #1: the act and sat test vocabulary in different ways.

traditionally, an “sat word” has been a vocabulary word that is both uncommon and difficult. you know, the sort of word you’d only get by drilling vocabulary into your head until your brain bleeds.

(author’s note: please don’t try that at home.)

the act, on the other hand, tests vocabulary that isn’t too terribly difficult. they’re words you would probably see in real life, and you’ll probably come across them in some of your reading in college — which you will have, regardless of your major. sorry.

what this means for you: you’re unlikely to be asked about weird, difficult vocabulary words, but you’ll be expected to know very detailed definitions of more common words. you also won’t be asked any straight-up vocabulary questions, but you will be expected to use words appropriately in the context given to you. so if you’re taking both tests, or if your school or teachers are offering ways to help you prepare for the sat, not everything applies from the sat to the act — though knowing more words can’t hurt!
 

top tip #2: the act loves secondary definitions, like whoa.

imagine with me for a moment. you’re in the middle of your act reading test, and you come across the word “suffer” in one of the passages. if you’re anything like me, when you read the word just now, you probably thought that someone was in pain or going through trauma of some kind.

but what if the sentence was something like, “she suffers from a tendency to exaggerate.” here, the woman in question isn’t in pain. i used a secondary definition of “suffer,” which here means “she is given to exaggeration” or, in more intelligible english, “she exaggerates a lot.”

or what if the sentence is something like the famous bible quote, “suffer the little children to come unto me”? don’t worry, no one is hurting kids. “suffer” is being used in yet another way: as a synonym for “allow” or “permit.” in this context, “suffer” means “allow the little children to come to me.”

what this means for you: if you think you know what a word means in an act question, you probably need to double-check the context. even if you don’t think you need to, do it anyway.

another thing you should double-check: parts of speech. if you’re dealing with the word “determined,” for example, you should be totally clear on whether the word is being used as an adjective (alyson was a determined young lady) or as the past tense of the word determine (galileo determined that the earth orbits the sun).
 

top tip #3: the act also loves idioms.

idioms are one of those lovely things about english that make the people learning it as a second language want to hit native speakers with the nearest lexicon until internal consistency is achieved. an idiom is a common phrase that makes no sense when you think about it, but we all know what it means in context. some familiar examples of idioms include:

narrow down — to reduce the number of choices or possibilities
cut corners — to do something poorly, often to save money
up in the air — to have undefined plans
up in arms — to become angry about something
hush-hush — secret or hidden
stumble upon — to discover accidentally
came about — happened

(if you’re looking for more practice with idioms for the act english test, we have a whole post on them here.)

what this means for you: now, the act won’t ask you to complete the sentence with the appropriate word — but idioms do appear fairly frequently in the reading test passages, so you should be prepared. if you come across an idiom and you’re not sure what it means, ignore it. if there’s a question based on that part of the passage, go back and give it a second look. try to predict what the author meant in that situation. chances are good that you’ll be able to get a general sense of the meaning, even if you’re not completely sure what all the words mean.

the post top tips for act vocab appeared first on magoosh blog | high school.

you may be thinking, though, that since you’re taking the act, not the sat, you don’t need to worry about vocabulary. you may even feel sorry for your classmates who are taking the sat; they have to deal with those excruciatingly difficult vocabulary questions. you may want to give a smug little laugh at the idea.

well, hold on, partner, because the act tests vocabulary, too.

now that i’ve sufficiently ruined your day, let’s talk about one of the ways that the act tests your vocabulary: jargon.

jargon is just a fancy word for specialized vocabulary. every profession, hobby, interest, or group has its own vocabulary that they don’t need to define within themselves every biologist will know the terms genome and punnett square; every decent seamstress knows her back stitch from her batting; every gamer can tell his rpg’s from his fps’s.

it makes talking to that group really difficult when you’re not part of it! they’ll use language that will have you scrambling for a gamer-to-english dictionary. and that’s jargon in a nutshell: vocabulary that the author didn’t need to define for his or her intended audience because they already knew what those words meant.

for you on the act, that means in nearly every section (i’m looking at you, science test!), there will be some vocabulary that is included just to intimidate you. jargon is intended to throw you off your game and use your fear of “sat words” against you. basically:

luckily, you don’t need a biology-to-english dictionary here! there is one surefire way to deal with jargon: deal with it as it comes. if the word or concept is important to the passage, it will be defined or revisited later. if there’s a question about that particular vocabulary word, you have my permission to puzzle over what it means. otherwise, don’t even pay attention to it. keep your cool, don’t be intimidated, and handle the jargon only if you need to.
 

the post tricksy little questionses: how the act tests vocabulary with jargon appeared first on magoosh blog | high school.