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

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

Let’s practice… 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

Make sure you have a calculator… Given Ratio of sidesAngle, side Looking for Angle measureMissing side Use SIN -1 COS -1 TAN -1 SIN, COS, TAN Set your calculator to ‘Degree’….. MODE (next to 2 nd button) Degree (third line down… highlight it) 2 nd Quit

Let’s practice… 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 TAN -1 (2/3) = 34°

Practice some more… Find tan A: A 21 Tan A = opp/adj = 12/21 Tan A = A Tan A = 8/4 = 2 8 Find tan A:

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

Using trig ratios in equations Remember back in 1 st grade when you had to solve: 12 = x What did you do? 6 (6) 72 = x Remember back in 3rd grade when x was in the denominator? 12 = 6 What did you do? x (x) 12x = 6 __ 12 x = 1/2

x cm 15 cm 34° Ask yourself: In relation to the angle, what pieces do I have? Opposite and hypotenuse Ask yourself: What trig ratio uses Opposite and Hypotenuse? SINE Set up the equation and solve: Sin 34 = x 15 (15) (15)Sin 34 = x 8.39 cm = x

x cm 12 cm 53° Ask yourself: In relation to the angle, what pieces do I have? Opposite and adjacent Ask yourself: What trig ratio uses Opposite and adjacent? tangent Set up the equation and solve: Tan 53 = x 12 (12) (12)tan 53 = x cm = x

x cm 18 cm 68° Ask yourself: In relation to the angle, what pieces do I have? Adjacent and hypotenuse Ask yourself: What trig ratio uses adjacent and hypotnuse? cosine Set up the equation and solve: Cos 68 = 18 x (x) (x)Cos 68 = 18 X = 18 cos 68 _____ cos 68 X = cm

42 cm 22 cm θ This time, you’re looking for theta. Ask yourself: In relation to the angle, what pieces do I have? Opposite and hypotenuse Ask yourself: What trig ratio uses opposite and hypotenuse? sine Set up the equation (remember you’re looking for theta): Sin θ = Remember to use the inverse function when you find theta THIS IS IMPORTANT!! Sin = θ °= θ

17 cm 22 cm θ You’re still looking for theta. Ask yourself: What trig ratio uses the parts I was given? tangent Set it up, solve it, tell me what you get. tan θ = THIS IS IMPORTANT!! tan = θ °= θ

Your assignment 3 Trig Worksheets (Kuta)