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.

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 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. opposite adjacent hypotenuse

B 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. opposite adjacent hypotenuse

Remember we won’t use the right angle X

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. One more thing…

Here we go!!!!

Trigonometric Ratios Name “say” SineCosinetangent Abbreviation Abbrev. SinCosTan Ratio of an angle measure Sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ =opposite side adjacent side

Values of Trigonometric Function Sine00.5 1/  2  3/2 1 Cosine1  3/21/  Tangent0 1/  3 1 33 Not defined CosecantNot defined2 222/  3 1 Secant1 2/  3 22 2Not defined CotangentNot defined 33 1 1/  3 0

One more time… Here are the ratios: One more time… Here are the ratios: sinθ = opposite side hypotenuse cosθ = adjacent side hypotenuse tanθ = opposite side adjacent side

B c a C b A Write the ratio for sin A Sin A = a c Write the ratio for cos A Cos A = b c Write the ratio for tan A Tan A = a b Let’s switch angles: Find the sin, cos and tan for Angle B: Sin B = b c Cos B = a c Tan B = b a

Set your calculator to ‘Degree’….. MODE (next to 2 nd button) Degree (third line down… highlight it) 2 nd Quit Calculator Commands

Find tan A: A 21 Tan A = opp/adj = 12/21 Tan A = A Tan A = 8/4 = 2 8 Find tan A:

Note: GivenLooking forUse Ratio of sides Angle measure SIN -1 COS -1 TAN -1 Angle, side Missing sideSIN, COS, TAN Set your calculator to ‘Degree’….. MODE (next to 2 nd button) Degree (third line down… highlight it) 2 nd Quit Calculator Commands Reminder

To solve for Angles: AB C xoxo Opp’ Adj’ Now we need to look at the two ratios involving the hypotenuse: sin x o = Opposite Hypotenuse hyp’ cos x o = Adjacent Hypotenuse

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. 14.8cm 9.7cm b o (1) Identify the two sides marked. a h (2) Choose the correct trig ratio. (3) Substitute in values. (4) Calculate the ratio(3 decimal places). (5) Use the inverse cosine function on your calculator to calculate the angle. b o = 49.1 o

Remembering the Trigonometric Ratios: Look again at the three trig ratios given below: Take the first letter of each word. Write the letters in order. SOHCAHTOA

C 2cm B 3cm A Find an angle that has a tangent (ratio) of 2 3 Round your answer to the nearest degree. 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

Class Work In your notebook, solve the following: CYU # 12, 13, 14, 15, 16 on pages

Work Period  Work with your group members about the final design of your pet house.  Remember : Your scale drawing must be accurate and precise.