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Math 10 Ms. Albarico Students are expected to: •Demonstrate an understanding of and apply properties to operations involving square roots. • Relate the trigonometric functions to the ratios in similar right triangles. • Use calculators to find trigonometric values of angles and angles to find when trigonometric values are known. * Solve problems using the trigonometric ratios. 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. We need to do some housekeeping before we can proceed… In trigonometry, the ratio we are talking about is the comparison of the sides of a RIGHT TRIANGLE. Two things MUST BE understood: 1. This is the hypotenuse.. This will ALWAYS be the hypotenuse 2. This is 90°… this makes the right triangle a right triangle…. Without it, we can not do this trig… we WILL NOT use it in our calculations because we COULD NOT do calculations without it. Now that we agree about the hypotenuse and right angle, there are only 4 things left; the 2 other angles and the 2 other sides. A We will refer to the sides in terms of their proximity to the angle hypotenuse adjacent opposite If we look at angle A, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse. If we look at angle B, there is one side that is adjacent to it and the other side is opposite from it, and of course we have the hypotenuse. hypotenuse opposite adjacent B 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. Here we go!!!! Trigonometric Ratios Name “say” Abbreviation Abbrev. Ratio of an angle measure Sine Cosine tangent Sin Cos Tan Sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ =opposite side adjacent side Values of Trigonometric Function 00 300 450 600 900 Sine 0 0.5 1/2 3/2 1 Cosine 1 3/2 1/2 0.5 0 Tangent 0 1/ 3 1 3 Not defined Cosecant Not defined 2 2 2/ 3 1 Secant 1 2 2 Not defined 1 1/ 3 0 2/ 3 Cotangent Not defined 3 One more time… Here are the ratios: sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ = opposite side adjacent side Write the ratio for sin A B Sin A = a c c Write the ratio for cos A a C b Cos A = b c A Write the ratio for tan A Let’s switch angles: Find the sin, cos and tan for Angle B: Sin B = b c Tan A = a b Cos B = a c Tan B = b a Calculator Commands Set your calculator to ‘Degree’….. MODE (next to 2nd button) Degree (third line down… highlight it) 2nd Quit Find tan A: Tan A = opp/adj = 12/21 24.19 A 12 Tan A = .5714 21 Find tan A: Tan A = 8/4 = 2 8 4 A Note: Given Ratio of sides Angle, side Looking for Use Angle measure SIN-1 COS-1 TAN-1 Missing side SIN, COS, TAN Calculator Commands Reminder Set your calculator to ‘Degree’….. MODE (next to 2nd button) Degree (third line down… highlight it) 2nd Quit To solve for Angles: tan x o opp adj C hyp’ Now we need to look at the two ratios involving the hypotenuse: sin xo = Opposite Hypotenuse cos xo = Adjacent Hypotenuse Opp’ xo A Adj’ B Calculator Commands For Trigonometric Inverse Functions: 1) Press 2ND, use SIN for SIN-1 COS for COS-1 TAN for TAN-1 Calculate the angle b o below. h 14.8cm bo 9.7cm (1) Identify the two sides marked. a adj cos x hyp (2) Choose the correct trig ratio . 9.7 o cos b 14.8 (3) Substitute in values . o cos b 0.655 o b o cos 1 0.655 b o = 49.1o (4) Calculate the ratio(3 decimal places). (5) Use the inverse cosine function on your calculator to calculate the angle . Remembering the Trigonometric Ratios: Look again at the three trig ratios given below: opp sin x hyp o adj cos x hyp o opp tan x adj o Take the first letter of each word. Write the letters in order. S O H C A H T O A Find an angle that has a tangent (ratio) of 2 C 2cm B 3 Round your answer to the nearest degree. 3cm A Process: I want to find an ANGLE. I was given the sides (ratio). Tangent is opp adj Solution: TAN-1(2/3) = 34° Ok… we’ve found side lengths, now let’s find angle measures. Refer to your table… what function will we use to find angle measures? SIN-1 COS-1 TAN-1 These are called INVERSE FUNCTIONS. Homework! In your notebook, CYU # 18, 19, 20, 21, 22, 24, and 25 on pages 239-240. Class Work In your notebook, solve the following: CYU # 12, 13, 14, 15, 16 on pages 236-237. Work Period Work with your group members about the final design of your pet house. Remember : Your scale drawing must be accurate and precise.