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 somehousekeeping before we can proceed…
In trigonometry, the ratio we are talkingabout is the comparison of the sides of aRIGHT 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. If we look at angle A, there is We will refer to the sides A in terms of their proximity one side that is adjacent to it and the other side is opposite to the angle from it, and of course we have the hypotenuse. hypotenuseadjacent opposite
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. hypotenuseopposite B adjacent
Trigonometric RatiosName Sine Cosine tangent“say”Abbreviation Sin Cos TanAbbrev.Ratio of an Sinθ = opposite side cosθ = adjacent side tanθ =opposite sideangle hypotenuse hypotenuse adjacent sidemeasure
sinθ = opposite side hypotenuseOne more One moretime… time…Here are the cosθ = adjacent side Here are theratios: ratios: hypotenuse tanθ =opposite side adjacent side
Let’s practice… Write the ratio for sin A B Sin A = a c c Write the ratio for cos A a Cos A = b c C b A Write the ratio for tan ALet’s switch angles: Tan A = aFind the sin, cos and btan for Angle B: Sin B = b Tan B = b Cos B = a c a c
Make sure you have a calculator… Given Ratio of sides Angle, sideLooking for Angle measure Missing side SIN-1 SIN, COS, TAN Use COS-1 TAN-1 Set your calculator to ‘Degree’….. MODE (next to 2nd button) Degree (third line down… highlight it) 2nd Quit
Let’s practice… Find an angle that has a tangent (ratio) of 2 C 32cm Round your answer to the nearest degree. Process: B 3cm A 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: Tan A = opp/adj = 12/21 24.19 12 Tan A = .5714 A 21 Find tan A: Tan A = 8/4 = 2 8 A 4
Ok… we’ve found side lengths, nowlet’s find angle measures.Refer to your table… what function willwe use to find angle measures? SIN-1 COS-1 These are TAN-1 called INVERSE FUNCTIONS