goal: know how to set up different trig ratios TRIGONOMETRIC RATIOS goal: know how to set up different trig ratios

Where to start… 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.

Trigonometric Ratios Sine (pronounced “sign”) Cosine Name “say” Sine (pronounced “sign”) Cosine (pronounced “co-sign) tangent (pronounced “tan-gent) Abbreviation Abbrev. Sin Cos Tan Ratio of an angle measure Sinθ = opposite side hypotenuse cosθ = adjacent side tanθ =opposite side adjacent side

hypotenuse hypotenuse opposite opposite adjacent adjacent

SOH CAH TOA O S H A C H O T A One more time… Here are the ratios: sinθ = opposite side hypotenuse H A C cosθ = adjacent side hypotenuse H O T tanθ =opposite side adjacent side A SOH CAH TOA

INE OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE We could ask for the trig functions of the angle Θ by using the definitions.  c b SOHCAHTOA SOHCAHTOA Θ INE a OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE

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! 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 So am I! Θ 3

Trigonometric Ratios Let ∆ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows. Side adjacent to A b cos A = = hypotenuse c Side opposite A a sin A = = hypotenuse c Side opposite A a tan A = = Side adjacent to A b

Let’s practice… M c a N b L Write the ratio for sin L Sin L= _a c Write the ratio for cos L Cos L = _b_ Write the ratio for tan L Tan L = _a_ b M c a N b L Let’s switch angles: Find the sin, cos and tan for Angle M: Tan M = _b_ a Sin M = _b_ c Cos M = _a_ c

65 a x Practice Together: Given each triangle, write the ratio that could be used to find x by connecting the angle and sides given. 32 b x

YOU DO: 56 d x Given the triangle, write all the ratios that could be used to find x by connecting the angle and sides given. c

Ex. 2: Finding Trig Ratios opposite sin S = hypotenuse adjacent cosS = hypotenuse opposite tanS = adjacent

Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle. opposite sin S = hypotenuse adjacent cosS = hypotenuse opposite tanS = adjacent

Ex. 1: Finding Trig Ratios Large Triangle Small Triangle opposite sin A = hypotenuse adjacent cosA = hypotenuse opposite tanA = adjacent

Ex. 3: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 45 opposite sin 45= hypotenuse adjacent cos 45= hypotenuse opposite tan 45= adjacent √2 45

Ex. 4: Finding Trig Ratios—Find the sine, the cosine, and the tangent of 30 opposite sin 30= hypotenuse adjacent cos 30= hypotenuse opposite tan 30= adjacent 30 √3