RIANGLE Trigonometric Ratios A RATIO is a comparison of two numbers. For example; boys to girls cats : dogs right : wrong. In Trigonometry, the comparison is between sides of a triangle. Finding Trig Ratios • A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. • The word trigonometry is derived from the ancient Greek language and means measurement of triangles. • The three basic trigonometric ratios are called sine, cosine, and tangent Finding Trig Ratios Some terminology: • Before we can use the ratios we need to get a few terms straight • The hypotenuse (hyp) is the longest side of the triangle – it never changes • The opposite (opp) is the side directly across from the angle you are considering • The adjacent (adj) is the side right beside the angle you are considering A picture always helps… • looking at the triangle in terms of angle b A is the adjacent (near the angle) C B is the opposite (across from the angle) C is always the hypotenuse b Longest A B hyp b Near adj opp Across But if we switch angles… • looking at the triangle in terms of angle a A is the opposite (across from the angle) C A a B is the adjacent (near the angle) C is always the hypotenuse B Across hyp Longest opp a adj Near Remember we won’t use the right angle X One more thing… θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’. Don’t let it scare you… it’s like ‘x’ except for angle measure… it’s a way for us to keep our variables understandable and organized. In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse A We’ll label them a, b, and c and the angles and . Trigonometric functions are defined by taking the ratios of sides of a right triangle. adjacent c leg b First let’s look at the three basic functions. SINE B leg a COSINE TANGENT They are abbreviated using their first 3 letters opposite a sin A hypotenuse c opposite a tan A adjacent b adjacent b cos A hypotenuse c the trig functions of the angle B using the definitions. A SOHCAHTOA c b adjacent a B opposite b sin B hypotenuse c adjacent a cos B hypotenuse c opposite b tan B adjacent a A It is important to note WHICH angle you are talking about when you find the value of the trig function. Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so c 5 4b adjacent a3 sin A = o 3 h 5 B tan B = o 4 3 a SOH-CAH-TOA You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle. Oh, I'm acute! A This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle. 5 4 3 B So am I! One more time… Here are the ratios: sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ =opposite side adjacent side The nice thing is that your calculator has a tan, sin and cos key that can save you some work. However, you must remember to change the settings on your calculator from Radians to Degrees. Radians is the default setting. Here are the steps: •Press Mode •Move the curser down to the 3rd line “Radians” •Slide the curser over so “degree” is blinking •Press 2nd Quit Make sure you have a calculator… Given Ratio of sides Angle, side Looking for Angle measure Missing side Use SIN-1 COS-1 TAN-1 SIN, COS, TAN How can we use these? Calculating a side if you know the angle you know an angle (25°) and its adjacent side we want to know the opposite side A tan 25 6 .47 A 1 6 6 (.47) A 2.80 A b C A 25° B=6 opp tan adj Another example • If you know an angle and its opposite side, you can find the adjacent side. opp tan adj 6 tan 25 B .47 6 1 B C b A=6 25° B .47 B 6 .47 B 6 12.76 .47 .47 B 12.76 How can we use it? Suppose we want to find an angle and we only know two side lengths Suppose we want to find angle a • Is side A opposite or adjacent? the opposite • what is side B? b C A=3 a B=4 the adjacent • with opposite and adjacent we use the… tan ratio opp tan adj Lets solve it b C opp tan adj 3 tan a 0.75 4 0.75 a tan a 36.87º A=3 a B=4 When the tan, sin or cos is in the denominator, we are going to use the reciprocal buttons. Look above tan on the calculator You should see TAN-1 Press 2nd TAN (.75) Another tangent example… • • • • we want to find angle b B is the opposite A is the adjacent so we use tan 4 tan b 3 tan b 1.33 b 53.13 opp tan adj b C A=3 a B=4 Ex. 6: Indirect Measurement • You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet. The math opposite tan 59° = Write the ratio adjacent h tan 59° = Substitute values 45 45 tan 59° = h Multiply each side by 45 45 (1.6643) ≈ h Use a calculator or table to find tan 59° 75.9 ≈ h Simplify The tree is about 76 feet tall. Ex. 7: Estimating Distance • Escalators. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg of 76 feet. d 30° 76 ft Now the math opposite sin 30° = hypotenuse 76 sin 30° = d Write the ratio for sine of 30° 30° Substitute values. d d sin 30° = 76 76 d= Divide each side by sin 30° sin 30° 76 d= Multiply each side by d. 0.5 d = 152 Substitute 0.5 for sin 30° Simplify A person travels 152 feet on the escalator stairs. 76 ft Why do we need the sin & cos? • We use sin and cos when we need to work with the hypotenuse • if you noticed, the tan formula does not have the hypotenuse in it. • so we need different formulas to do this work • sin and cos are the ones! C = 10 b A 25° B Lets do sin first • we want to find angle a • since we have opp and hyp we use sin 5 sin a 10 sin a 0.5 a 30 opp sin hyp C = 10 b A=5 a B And one more sin example • find the length of side A • We have the angle and the hyp, and we need the opp opp sin hyp A sin 25 20 A sin 25 20 A 0.42 20 A 8.45 C = 20 b A 25° B And finally cos • We use cos when we need to work with the hyp and adj adj cos • so lets find angle b hyp 4 cos b 10 cos b 0.4 b 66.42 C = 10 b A=4 a a 90 - 66.42 a 23.58 B Here is an example • Spike wants to ride down a steel beam • The beam is 5m long and is leaning against a tree at an angle of 65° to the ground • His friends want to find out how high up in the air he is when he starts so they can put add it to the doctors report at the hospital • How high up is he? How do we know which formula to use??? • • • • • Well, what are we working with? We have an angle We have hyp We need opp With these things we will use B the sin formula C=5 65° So lets calculate opp sin 65 hyp opp sin 65 5 opp sin 65 5 opp 0.91 5 opp 4.53 • so Spike will have fallen 4.53m C=5 B 65° One last example… • Lucretia drops her walkman off the Leaning Tower of Pisa when she visits Italy • It falls to the ground 2 meters from the base of the tower • If the tower is at an angle of 88° to the ground, how far did it fall? First draw a triangle • • • • • What parts do we have? We have an angle We have the Adjacent We need the opposite Since we are working with the adj and opp, we will use the tan formula B 88° 2m So lets calculate opp tan 88 adj opp tan 88 2 opp tan 88 2 opp 28.64 2 opp 57.27 • Lucretia’s walkman fell 57.27m B 88° 2m An application You look up at an angle of 65° at the top of a tree that is 10m away the distance to the tree is the adjacent side the height of the tree is the opposite side opp tan 65 10 opp 10 tan 65 opp 10 2.14 opp 21.4 65° 10m What are the steps for doing one of these questions? 1. 2. 3. 4. 5. 6. 7. Make a diagram if needed Determine which angle you are working with Label the sides you are working with Decide which formula fits the sides Substitute the values into the formula Solve the equation for the unknown value Does the answer make sense? Two Triangle Problems • Although there are two triangles, you only need to solve one at a time • The big thing is to analyze the system to understand what you are being given • Consider the following problem: • You are standing on the roof of one building looking at another building, and need to find the height of both buildings. Draw a diagram • You can measure the angle 40° down to the base of other building and up 60° to the top as well. You know the distance between the two buildings is 45m 60° 40° 45m Break the problem into two triangles. • The first triangle: a • The second triangle • note that they share a side 45m long • a and b are heights! 60° 45m 40° b The First Triangle • We are dealing with an angle, the opposite and the adjacent • this gives us Tan a tan 60 45 a tan 60 45 a 1.73 45 a 77.94m a 60° 45m The second triangle • We are dealing with an angle, the opposite and the adjacent • this gives us Tan b tan 40 45 b tan 40 45 b 0.84 45 b 37.76m 45m 40° b What does it mean? • Look at the diagram now: • the short building is 37.76m tall • the tall building is 77.94m plus 37.76m tall, which equals 115.70m tall 77.94m 60° 40° 37.76m 45m Ex: 5 Using a Calculator • You can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.