Chapter 15 Right Triangle Trigonometry Sec. 1 Right Triangle Trigonometry The most difficult part of Trigonometry is spelling it. Once we get by that, the rest is a piece of cake. efore we start naming the trigonometric ratios, let s look at a right triangle. c a b C Using the sides; a, b, c, how many fractions can you make? Let s see: We have a c, b c, b a, c a, c b, a b There are a total of six different fractions. Since those fractions are comparing 2 sides of a triangle, we will call those RTIOS. So we have six different ratios. Now, how many trigonometric ratios do you think there might be? If you answered six, you are right. What we do is give those ratios names; we call them sine, cosine, tangent, cosecant, secant, and cotangent. Now I don t know who thought those names up, but we do have to know them. Let s look at that triangle again. c a b C Page 1
From your geometry, you might remember, the side opposite the right angle is called the hypotenuse. So side c is always the hypotenuse. Now, if we stand on a is opposite that and b is adjacent to. Now stay with me, if we stand on, what side is opposite? Yes, b. The hypotenuse does not change, that is still c. What side is adjacent to? ll right, it is a. The way we are going to name those trig ratios is in terms of where those sides are located with respect to the angle we are working with (standing on). If we are standing on, what side is opposite? If we are standing on, what side is opposite? Notice the opposite side will be determined by where you are standing. Now the way we name the trig ratios is by using the terms: hypotenuse, opposite, and adjacent. I want to give three of those ratios we looked at names. The sine of an angle is defined to be the opposite side over the hypotenuse. Symbolically stated: Sine of an angle is opposite hypotenuse Cosine of an angle is Tangent of an angle is opposite adjacent adjacent hypotenuse We typically abbreviate those names: sin, cos, and tan for sine, cosine, and tangent. Now, we ll look at another triangle using numbers instead of letters and find the sin, cos, and tan. You will love this. Math is your life. To help you remember the formulas for those three trig ratios, I will introduce you to SOHCHTO. SOHCHTO lives in a lean-to. Page 2
s you can see, his lean-to looks like a right triangle. We are not going to name his horse or what kind of tree is in the picture. ut SOHCHTO s name has some significance. Can you see it? OK, I ll tell you. S stands for sine, O for opposite, H for hypotenuse, for adjacent, C for cosine, and T for tangent. Therefore, Sine is the opposite over the Hypotenuse. Cosine is the djacent over the Hypotenuse, and the Tangent is the Opposite over the djacent. Don t you just love this? Example 1 Let s find the sin, cos, and the tan 5 3 4 C Remember SOHCHTO. sin = 3/5 cos = 4/5 tan = 3/4 Was that hard? Of course not. Using the same triangle, find the sin, cos, and tan. sin = 4/5 cos = 3/5 tan = 4/3 Make sure you spell SOHCHTO correctly, that way you will know the trig ratios. s you can see, this isn t too bad. We ll do more next time, but try this one for now. Page 3
Example 2 Find the sin, cos, and tan for angles Q and R. Q 13 12 R S 5 Using SOHCHTO sin Q = 5/13 sin R = 12/13 cos Q = 12/13 cos R = 5/15 tan Q = 5/12 tan R = 12/5 Just as a heads-up as you move forward in your study of trig, Look at the relationships(ratios) of those angles. nd know those relationships seem to occur when the two angles are complementary. Sec 2. Problem solving with trig ratios Last time we looked at a right triangle, we found all the possible fractions that could be made by using the sides of the triangle. We called those fractions ratios, then we gave those six ratios special names; sine. cosine, tangent, cosecant, secant, and cotangent. Now, the question that arises is what happens if we have a different size right triangle? We are going to have to recall some geometry, in particular SIMILR triangles. Two triangles are SIMILR if the corresponding angles are equal and all the corresponding sides are in proportion. Yes, I am absolutely sure you remember that. One theorem that I am also sure you will remember about similar triangles is: Page 4
Theorem If a line is drawn parallel to one of the sides of a triangle, then the two triangles formed are similar. That is the corresponding angles are equal and the corresponding sides are in proportion. Let s look. X Y C If I draw XY anywhere so it is parallel to C, then we have two triangles, C and XY. y using the previous theorem, C is similar to XY. That means angle of the small triangle equals angle of the large triangle, angle X of the small triangle equals angle, and by the same reasoning, angle Y equals angle C. It also means the sides are in proportion. That is, when I make the ratios for the small triangle using the sides, those corresponding ratios for the large triangle will be the same. Example 2 Write 2 relationships based on the triangles being similar. Y XY = C C, XY X = C These ratios will NOT change unless the angle changes. The consistency allows to give these ratios the name sine, cosine, tangent, etc. Making that a little clearer; if the angle changes, then the ratio will also change. Putting that more succinctly, the trigonometric measure of the angle of a triangle will have it s own unique ratio. That means, the sine of a 30 degree angle, in a large or small triangle will be 1/2.. Remember the sine is the opposite over the hypotenuse. SOHCHTO. Page 5
30 2 3 1 That constancy will allow us to solve problems using trigonometry. y using many different right triangles, we are able to find the ratios of the sides for different angles, We then are able to make what is known as the TRIGONOMETRIC TLES. Those tables allow us to look up those ratios given the associated angles. Now that s neat! Use the tables in your calculator or in your book to find the sine, cosine and tangent of the following angles. 1. 30 2. 60 3. 40 4. 63 Find the measue of the angle with the following values. 5. Sin =.5150 6. Cos =.7071 7 tanc = 1.3270 To use trig to solve problems, we have to know those trig ratios. I find it easy to remember them by recalling SOCHCHTO. Example 3 boy standing on level ground notices (or can measure) that the angle he has to look up (angle of elevation) to see the top of a flagpole is 42 degrees. He can also measure the distance he is from the pole and finds it to be 120 feet. How high is the flagpole? Page 6
We ll call the height of the flagpole h. Let s look at a picture and fill in any information we have 42 120 h Now, how can we determine the height? Well, since we are studying trig, let s hope we use one of the trig ratios. Which one do we use? That s the question. The sine is the opposite over the hypotenuse; we don t know the opposite or the hypotenuse. So we won t use the sine. Cosine is the adjacent over the hypotenuse; that does not give the height. So I won t use that. Tangent is the opposite over the adjacent. There, I know the adjacent is 120 and I m looking for the opposite (height). Therefore, I will use the tangent ratio. I will have to look up the tan 42 on my trig table. The tan 42 =.9004 tan 42 = opposite adjacent = h 120.9004 = h 120 so h is approximately 108 feet To solve problems involving the trigonometric ratios, it s always wise to think first, then set up a ratio. s we have said before, the choices we make can either make easier or more cumbersome. So, we have to ask, given crtain information, am I going to use the sine cosine or tangent and from what angle? So thinking first is important. Page 7
Example 4 Find the measure of DE in the diagram. F D 32 x 200 E Using the trig ratios, we are working with 2 sides and an angle. To solve problems, we need to know 2 of those 3 and find the unknown. In this problem, we have 2 sides and an angle. Notice that the hypotenuse is NOT one of the sides. For me, that eliminates using the sine or cosine ratios. So, that leaves me with the tangent ratio, do I want to use angle D or F? Remember, if D = 32, then F = 58. The sum must be 90 So now decide, which is an easier equation to solve, or?. tan 32 = 200 x or. tan58 = x 200 In either case< I have to use a table or calculator to find the respective tangent values. However, since x is in the numerator of the., I would choose to multiply, rathe then divide. The tan 58 1.6003, tan58 = x 200 x 200(1.6003) 320 Page 8
In a number of applications, professionals use terms like angle of elevation or angle of depression angle of depression angle of elevation Line segment is the line of sight, the other two lines are horizontals. Let s examine what we know. ll trig is in the study of the ratios of the sides of the right triangle. Those ratios are given names; sine, cosine, tangent, cosecant, secant, and cotangent. To know which ratio we are talking about, we stand on the vertex of the angle and use the acronym SOHCHTO. y looking at different right triangles and knowing the ratios are the same for equal angles (using SIMILR TRINGLES) we are able to make trig tables for all the angles that will give us the ratio of those sides. Don t you just love this stuff? You are probably thinking trig is my life. 1. The length of a kite string is 100 feet long. how high is the kite if the kite make a 30 angle with the round? 2. From the top of a building 300 feet high, you see a parked car on a bridge, if the angle of depression is 60, how far is the car from the building? 3. pilot reads the angle of depression from his position 5000 feet in the air is 20, how far is the plane from the airport? Page 9