Trig for right triangles is pretty straightforward. The three, basic trig functions are just relationships between angles and sides of the triangle.

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1 Lesson 10-1: 1: Right ngle Trig By this point, you ve probably had some basic trigonometry in either algebra 1 or geometry, but we re going to hash through the basics again. If it s all review, just think of it as a chapter you get to breeze through. Trig for right triangles is pretty straightforward. The three, basic trig functions are just relationships between angles and sides of the triangle. sin θ = opposite hypoteneuse cos θ = adjacent hypoteneuse tan θ = opposite adjacent Now, you may be saying, What is that little crossed out oval thing? What s θ? That s the Greek letter, theta. Mathematicians often use it (or phi, which you may run into later) to represent angles. I guess they got tired of Latin letters or something. Nobody expects you to know the trig functions of most angles off the top of your head. They used to look them up in tables; now we have calculators for that sort of thing. Just be absolutely sure that your calculator is in degree mode. It will probably say DEG or something on your screen, but every calculator is different. If you don t see anything, find the tangent of 45. If your answer is 1, you re in good shape. If it s you re in the wrong mode, and you d better fix it real soon! Ex. 1: Find the values of sine, cosine, and tangent nt for the given angle: θ 5 7 θ 4

2 Your textbook wants you to memorize the functions for two, special triangles. Feel free to do so. I never do. Instead, I memorize the following table, which has such a sweet pattern to it that I find it hard to forget: 0 o 30 o 45 o 60 o 90 o Now, you ll need to do some sin simplifying each time. fter all, you can t have radicals in the cos denominator,, but most times, that s pretty simple. tan The other issue is the tangent of 90 o. That s a wee bit of a problem because the square root of 0 is 0, and then you re dividing by zero, which is a terrible, terrible mistake. Since you can t divide by zero, tan 90 o is undefined. Ex. : Solve x x o Trig functions can also be used to find angles of elevation and angles of depression.. The first is what angle something is off the ground (or straight horizontally from the observer) while the second is the angle something is below the observer. ngle of depression ngle of elevation

3 Ex. 3. biologist whose eyes are at 6 ft exactly measures the angle of elevation to the top of a tree to be 38.7 o. If the biologist is standing 180 ft. from the tree s base, what is the height of the tree? It became handy to develop reciprocal trig functions for certain situations as well. They are: 1 hyp. cscθ = = sinθ opp. secθ = 1 cos hyp. = θ adj. cotθ = Ex. 4: Find the values for all 6 trig functions: 1 tan adj. = θ opp θ

4 Lesson 10-: ngles of Rotation Trig isn t just for right triangles. If it were, it would get real useless real fast. It s all a bunch of angles, so let s drop an angle on the coordinate plane. You can tilt that plane however you want (or you can tilt the angle however you want) to make it right at the origin with one line lying along the positive x-axis. This is standard position.. The line on the x-axis is the initial side and the other is the terminal side. Terminal Side II I Initial Side Vertex III IV You create an angle of rotation by spinning the terminal side around the origin.. Counter-clockwise rotation is considered positive, and clockwise is considered negative. You don t necessarily, of course, need to keep your terminal side in quadrant I.. It could end up anywhere. Ex. 1: Draw the following angles in standard position: a. 30 o b. 110 o c. 990 o Coterminal angles are angles in standard position whose terminal sides are equal. Doesn t matter how they got there, just so long as they t get there. Ex. : Find the measure of two coterminal angles of 65 o

5 So I m missing what this has to do with trig. I know; I know! I m getting there! Let s sketch an angle in quadrant I. dd a little dotted line (this will be the only time we do this), and voila! You have a right triangle! Now let s try that in quadrant II. dd a little dotted line (okay, I lied. This will be the only OTHER time we do this. In the future, it s completely unnecessary.), and now we have another right triangle. Of course, that triangle doesn t use the same angle as the other one. It uses the reference angle, the angle formed by the shorted path from the terminal side to the x-axis (always x, never y). Ex. 3: Find the reference angle for the following: a. 135 o b. 105 o c. 35 o Now any time you stick a point on a coordinate plane, you can draw a line to it from the origin and find trig functions related to the angle created from the positive x-axis. Ex. 4: If P(-6, 9) is a point on the terminal side of θ in standard position, find the exact value of the six trig functions for θ.

6 Lesson 10-3: The Unit Circle Earlier, I made sure you were in degree mode on your calculator. The question left unanswered is what other options are there? Simple: radians. Radians are measures of angles based on arc length instead of an arbitrary system of 360 o. Suppose you have a circle of radius 1 (the unit circle). n angle that slices out an arc of length 1 has a measure of 1 radian. If your radius is 1, what s the circumference of the entire circle? C = πr Plug in 1 for r, and you get a circle with h a circumference of π.. That means the radian measure of the entire circle is also π. Thus, π radians = 360 o (though in practice, radians have no actual unit) Ex. 1: Convert each measure from degrees to radians or back. a. 60 o π b. 3 Now lay that unit circle down on a coordinate system along with an angle. The spot where the angle intersects with the circle has coordinates x & y. 1 cos θ = x/1 = x y θ sin θ = y/1 = y This means that any point can also be described as (cos θ,, sin θ). x

7 Since the unit circle is so useful in finding x and y via trig, sine and cosine are also referred to as circular functions. The rest of the trig functions have their own relationships to the unit circle: tan θ cot θ sec θ csc θ Ex : Use the unit circle to find the value of the following: a) sin (-90( ) b) cot (180 o ) c) csc (3π/) y x x y 1 x 1 y In the previous lesson, we introduced reference angles. Why? Because reference angles can be used to find the trig function of any angle pretty easily: 1. Find the reference angle. ct like the reference angle is in quadrant I 3. Change the sign of the result based on the quadrant it s actually in How do you know what the sign is supposed to be? Well, if sin θ is y and cos θ is x sin cos tan sin cos tan sin cos tan sin cos tan

8 Ex. 3: Use a reference angle to find the sine, cosine, and tangent of 330 o. If you know an angle, you should be able to find the length of the arc it creates in not only a unit circle, but any circle. So long as your angle is measured in radians, just multiply by the radius. It s that easy: s = rθ r Ex. 4: tire t on a car makes 653 complete rotations in 1 minute. The diameter of tire is 0.65 m. To the nearest meter, how far does the car travel in 1 second?

9 Lesson 10-4: Inverse Trigonometric Functions If you can do it, you d better be able to undo it. That s what inverse trigonometric functions are for. For example: If sin 90 o = 1, then sin -1 1 = 90 o or if π cos = 4, then cos 1 π = 4 Of course,, bear in mind that Note: Inverse trig functions 90 o o = 450 o, and that s just once are also sometimes known around the circle. That means that sin as arcsin, arccos, and 450 o = 1, which means that sin -1 1 = 450 o. arctan Can sin -1 1 = both 90 o and 450 o? Yup. When you re taking inverse trig functions, you get multiple answers. Be aware of what they all might be by paying attention to what quadrant the reference angle is in. Ex. 1: Find all possible values of cos 1 3. Express answers in radians. Because the inverses give you multiple results, mathematicians use principal values to note a particular set of values to be used for the domains of sine, cosine, and tangent. If you re only taking the principal values of sin, cos, and tangent, you just use capital letters The principal values for Sin, Cos, and Tan are Sin: -π/ < x < π/ Cos: 0 < x < π Tan: -π/ < x < π/ Therefore, the ranges for rcsin, rccos, and rctan (Sin -1, Cos -1, and Tan -1 ) are limited to those values as well.

10 Ex. : Evaluate each inverse. Give your answers in both radians and degrees. a Cos b. Sin Inverse trig functions allow you to solve various kinds of situations. Ex. 3: painter needs to lean a 30 ft. ladder against a wall. Safety guidelines recommend that the distance between the base of the ladder and the t wall should be 1/4 of the length of the ladder. To the nearest degree, what acute angle should the ladder make with the ground? Ex. 4: Solve sin θ = 0.4 for 90 o < θ < 70 o

11 Lesson 10-5: The Law of Sines So far, all we ve really done is work with right triangles (even if some of them are implicitly laid across circles). But trig can be used for *any* triangle, just with a little effort. Recall the formula for the area of f a triangle is rea = base. height/. That s easy for right triangles; the base and height are the two sides. For other triangles, you need to figure out the height. Just sketch a line from the point to the base. b c But now, you might notice, you have a couple of right B a triangles! Trig comes into play here. The height is the side of a right triangle with hypotenuse c. That means sin B = h c or h = c sin B. So the new formula is: rea = 0.5 a c sin B C Depending on how you label the triangle, it could be any of these: rea = 0.5 a c sin B rea = 0.5 a b sin C rea = 0.5 b c sin Ex. 1: Find the area of the triangle: o So those three formulas we have They ll all give you the same area. So they re all equal. So what happens when you set them equal and simplify? 1 acsin B = 1 absin C = 1 bcsin = Multiply by acsin B absin C = bcsin acsin B abc absin C abc bcsin = abc = Divide by abc sin B b sin C c sin = a = Simplify

12 That s the Law of Sines. You can use it to solve any triangle, right or not, so long as you have two angles and a side (S or S) or two sides and an angle not between them (SS). Ex. : Solve the triangle (obviously not to scale) 36 o o

13 Lesson 10-5B: The Law of Sines: The mbiguous Case There is a small problem with the Law of Sines. Sometimes, when you have a SS situation, you can end up with the ambiguous case.. This means that you might be able to use the information to build one triangle, two triangles, or no triangle at all! There are diagrams below, but I ll show on the board as well why this happens. To determine how many triangles you have, you need the following: The known angle (we ll call it, here) The corresponding side (we ll call it a) The other side (we ll call it b) and sometimes The height of the triangle, found using: b sin and the table shown below. Don t get confused if the names get mismatched. You can always rename the sides and angles of a triangle. Just make sure the angles and sides match. The mbiguous Case Table If > 90 o b a a b No solution b a a > b One solution a < b b No solution If < 90 o a < b sin θ a = b sin θ a > b sin θ a b a b a a b sin θ b sin θ b sin θ One solution Two solutions

14 a b b a One solution This is why the Law of Sines is actually a bit obnoxious at times If it turns out there are two possible triangles, this means that angle B, which you ll be finding first, has two possibilities, one from quadrant I (which your calculator will spit out) and one from quadrant II (180 - B). So any time you have SS, you must 1. Determine the number of possible solutions. If none, you re finished 3. If one, use the Law of Sines to solve the triangle 4. If two o Use the Law of Sines to solve for the first triangle o Go back and find the OTHER angle B (180 B 1 ) o Use this new angle to solve the rest of the new triangle Ex. 1: What triangles could be formed when = 4 o, a = 70, and b = 1?

15 Ex : Determine the complete dimensions of all the triangular banners that can be formed using the measurements a = 1, b =, and = 7. o Ex 3: Solve all triangles if B = 110 o, a = 105, and b = 70

16 Lesson 10-6: The Law of Cosines There s a Law of Cosines, too. Good news! There s no ambiguous case. Suppose we have this triangle. Drop a line to put the height in, and you end up with two right triangles. If the base of the original triangle is b, we ll call the base of one of the right triangles x, which makes the other b x. a b c Let s play with the Pythagorean theorem for those triangles we created: c = ( b x) + h Right triangle on the right a = x + h or h = a x Right triangle on the left c = ( b x) + a x Plugging in the nd formula into the 1 st c = b bx + x + a x Expanding (b x) c = a + b bx Simplifying and rearranging cos C = x or x = a cosc From the left triangle c a = a + b abcosc Plugging in from previous formula Plugging in from previous formula nd that, ladies and gentlemen, is the Law of Cosines. Or one form of it. You re hopefully sharp enough to figure out the other two. (a = ) nywhere the Law of Sines doesn t t work, this will. nd sometimes even where the Law of Sines does work, this will still work. It s also worth noting that once you ve used the Law of Cosines, no ambiguous cases can show up, even if you end up using the Law of Sines later. Ex. 1: Solve the triangle: a = 8, b = 9, c = 7

17 Heron s formula is based on the Law of Cosines and is used to find the area of a triangle. The proof is left to the interested student. (aka. I m either too lazy to bother with it or we re running too low on time). rea = s( s a)( s b)( s c) where s is the semiperimeter s = 1 ( a + b + c ) Ex. : garden has a triangular flower bed with sides measuring yd., 6 yd., and 7 yd. What is the area of the flower bed?