{"id":941,"date":"2015-07-23T09:00:01","date_gmt":"2015-07-23T16:00:01","guid":{"rendered":"https:\/\/magoosh.com\/act\/?p=941"},"modified":"2017-09-28T09:08:13","modified_gmt":"2017-09-28T16:08:13","slug":"act-trigonometry","status":"publish","type":"post","link":"https:\/\/magoosh.com\/act\/act-trigonometry\/","title":{"rendered":"ACT Math Trigonometry"},"content":{"rendered":"

Ah, ACT Trigonometry. I can hear your reactions from here, my lovely Magooshers. \u201cOh, wow, trigonometry is one of the ACT math topics<\/a> tested? Let me do my best cheerleader cheer! Cosine, secant, tangent, sine, 3.14159! Goooo Trig!<\/strong><\/em> Woo-hoo!\u201d<\/p>\n

Okay, I know most<\/em> of you aren\u2019t reacting that way (…yet), but I promise you\u2019ll quake in fear just a little less when this is all over.<\/p>\n

Trig is definitely the most intimidating math to most ACT students \u2013 that\u2019s because most of them haven\u2019t 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\u2019ll be all set to tackle even the hardest problems!<\/p>\n

Table of Contents<\/h2>\n
    \n
  1. First of All: What Is Trigonometry?<\/a><\/li>\n
  2. The Basics of Trig for ACT Math<\/a><\/li>\n
  3. Practice with SOHCAHTOA on ACT Math<\/a><\/li>\n
  4. Reciprocal Trig Identities<\/a><\/li>\n
  5. Some Helpful Hints for ACT Math Trig<\/a><\/li>\n
  6. Practice Makes Perfect<\/a><\/li>\n
  7. Graphing Sine, Cosine, and Tangent<\/a><\/li>\n
  8. Sine, Cosine, and Tangent Graphs in Practice: Transformations<\/a><\/li>\n
  9. The Unit Circle<\/a><\/li>\n
  10. ACT Unit Circle Practice<\/a><\/li>\n<\/ol>\n

    Note: for more helpful math formulas, including ones for trigonometry, look here<\/a> <\/em>
    \n    <\/a><\/p>\n

    First of All: What Is Trigonometry?<\/h2>\n

    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. <\/strong>If you like right triangles, you\u2019re going to do well here.<\/p>\n

    And typically the first thing you study in a trig class are right triangles:<\/p>\n

    \"Screen<\/p>\n

    So here\u2019s a right triangle. Let\u2019s 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.
    \n    <\/a><\/p>\n

    The Basics of Trig for ACT Math<\/h2>\n

    To help illustrate my next point, let me tell you a brief story.<\/p>\n

    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.<\/p>\n

    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\u2019t 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, \u201cYou must SOHCAHTOA.\u201d<\/p>\n

    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:<\/p>\n

    Sine = Opposite \/ Hypotenuse (SOH)<\/p>\n

    Cosine = Adjacent \/ Hypotenuse (CAH)<\/p>\n

    Tangent = Opposite \/ Adjacent (TOA)<\/p>\n

    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.<\/em> Remember to keep things from the right point of view. Opposite always means \u201copposite to the angle you\u2019re being asked about\u201d and adjacent always means \u201cnext to the angle you\u2019re being asked about.\u201d<\/p>\n

    <\/a><\/p>\n

    Practice with SOHCAHTOA for ACT Math<\/h2>\n

    Example 1: <\/strong><\/p>\n

    What is the sin of A?<\/p>\n

    \"Screen<\/p>\n

    Knowing SOHCAHTOA, you would be able to answer that it is opposite\/hypotenuse or \"3\/5\" . Easy as that!
    \nExample 2:<\/strong><\/p>\n

    What is the length of XZ?<\/p>\n

    \"Screen<\/p>\n

    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.
    \nSo in this case we can use sine to find the length of the hypotenuse.<\/p>\n

    sin(10) = 3 \/ XZ
    \nXZ = 3 \/ sin (10)<\/p>\n

    We can divide sin of 10 degrees by 3 in our calculator to get the answer: approximately 17.28.
    \nExample 3<\/strong><\/p>\n

    Here\u2019s one that\u2019s just a teensy bit harder, but we are just going to apply the same principles.<\/p>\n

    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 \"4\/5\". What is the height of the tree?<\/p>\n

    \"Screen<\/p>\n

    The problem tells us that the cosine of the angle of elevation is \"4\/5\". Remember SOHCAHTOA, so we are concerned with the adjacent side over the hypotenuse. The fact that the cosine is \"4\/5\" means the ratio of the adjacent side to the hypotenuse is \"4\/5\". So we can set up a proportion:<\/p>\n

    \"4\/5\" = \"24\/x\"<\/p>\n

    Cross-multiplying to solve for x gives us x = 30.<\/p>\n

    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.<\/p>\n

    \"24^2\" + \"b^2\" = \"30^2\"<\/p>\n

    =18<\/p>\n

    So the height of the tree is 18 ft.<\/p>\n

    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.
    \n    <\/a><\/p>\n

    Reciprocal Trig Identities<\/h2>\n

    You will definitely encounter questions that require you to use SOHCAHTOA, and you may encounter questions that ask about reciprocal trig identities<\/b>. <\/b>Each of the three basic trig identities has a corresponding reciprocal trig identity:<\/p>\n

    Cosecant = Hypotenuse \/ Opposite<\/b><\/p>\n

    Secant = Hypotenuse \/ Adjacent<\/b><\/p>\n

    Cotangent = Adjacent \/ Opposite<\/b><\/p>\n

    Notice how Sine and Cosecant are the same except the numerator and denominator is flipped. That\u2019s what we mean by reciprocal. It\u2019s easy to remember that \u201ctangent\u201d and \u201ccotangent\u201d are reciprocals since they sound so much alike, but how what about the other two? I once had a math teacher who used, \u201cCo-co no go\u201d<\/i> as a mnemonic device to help my high school class remember. What he meant was that your brain may think that \u201ccosine\u201d and \u201ccosecant\u201d are reciprocals since they both begin with the prefix \u201cco-\u201c but that isn\u2019t true. \u201cSine\u201d goes with \u201ccosecant\u201d and \u201ccosine\u201d goes with \u201csecant.\u201d<\/p>\n

    <\/a><\/p>\n

    Some Helpful Hints for ACT Math Trig<\/h2>\n

    Let\u2019s revisit our lovely little triangle:<\/p>\n

    \"at_img1a\"<\/p>\n

    You should know that you can do this:<\/p>\n

    \"at_img4\"<\/p>\n

    This is what\u2019s called the law of sines.<\/strong> Usually, if you have to use this formula, the question will give it to you, but it\u2019s a handy tool to have in your pocket.<\/p>\n

    Next up is a nifty little equation that you can use on any angle.<\/strong> We\u2019ll follow mathematical convention here and use the symbol \u03b8 (pronounced \u201ctheta\u201d) to stand in for the value of the angle.<\/p>\n

    \"at_img5\"<\/p>\n

    To translate from math back into English, the sine of any angle, squared, plus the cosine of any angle, squared, equals 1.<\/strong> Could be useful if you\u2019re 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\u2019s equal to 1.<\/p>\n

    And to round out our helpful hints, here\u2019s one last equation for you:<\/p>\n

    \"at_img6\"<\/p>\n

    Translation: The tangent of any angle equals the sine of the angle divided by the cosine.<\/strong> 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!<\/p>\n

    To recap, let\u2019s look at the equations all in one place:<\/p>\n

    \"a_t_img1\"<\/p>\n

    \"a_t_img2\"<\/p>\n

    Finally, an unusual definition to learn! A radian <\/b>is another way of measuring an angle. Some harder problems on the ACT will use radians instead of degrees.<\/p>\n

    There are 2\u03c0 radians in one circle. Each point on a circle corresponds to a certain number of radians.<\/p>\n

    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 3\u03c0\/2 and 2\u03c0, we know that the angle must be in the 4th<\/sup> quadrant of the circle.To convert degrees to radians, simply multiply by \u03c0\/180.<\/p>\n

    <\/a><\/p>\n

    Practice Makes Perfect<\/h2>\n

    With all of this in mind, let\u2019s do a sample problem!<\/p>\n

    \"at_img7\"<\/p>\n

    \"at_img8\"<\/p>\n

    The correct answer is\u2026 D! Let\u2019s walk through it.<\/p>\n

    To find the sine of \u2220A, 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\u2019t give you a value for line BC. It did<\/em> give you line AB, though, which is 6. So we can use the Pythagorean Theorem to figure out line BC!<\/p>\n

    \"{a^2} is the Pythagorean Theorem, as you might recall from the review on triangles. Substitute in the values we know, and it becomes:<\/p>\n

    \"at_img9\"<\/p>\n

    Now that we know the value of line BC, we can figure out the sine of \u2220A.<\/p>\n

    \"at_img10\"<\/p>\n

    And we have our answer!
    \n    <\/a><\/p>\n

    Graphing Sine, Cosine, and Tangent<\/h2>\n

    These graphs are usually graphed and expressed in degrees, but you may also see them expressed in radians.<\/p>\n

    Sine<\/b> and cosine<\/b> both have standard graphs that you need to memorize for the ACT Math Test<\/a>. The standard equation for sine looks like this: y = sin x<\/b>. The \u201cperiod\u201d of the wave is how long it takes the curve to reach its beginning point again. The coefficient in front of \u201csin\u201d (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.<\/p>\n

    \"agt_img1\"<\/a><\/p>\n

    Cosine also has a standard equation. It looks like: y = cos x<\/b>. 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 2\u03c0. 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 ?<\/i><\/p>\n

    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 \u03c0\/2. For the new graph, it will occur at \u03c0\/4, which is \u00bd of \u03c0\/2.<\/p>\n

    Confused? Want more? Kristin’s here to explain!<\/p>\n