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8-3 8-4 Sine, Cosine and Tangent Ratios Objective Students will be able to use sine, cosine, and tangent ratios to determine side lengths in triangles. Trigonometric Ratios • We use the Pythagorean Theorem when we are given two sides of a right triangle and we want to find the third. • We will use trigonometric ratios when we are given one side and one angle (other than the 90 degree angle) of a right triangle and want to find another (or both) side(s). • There are three trigonometric ratios we will utilize • Sine (abbreviated sin) • Cosine (abbreviated cos) • Tangent (abbreviated tan) Side Definitions • Opposite side: leg directly across from the angle of interest • Adjacent side: leg next to the angle of interest • Hypotenuse: side directly across from the right angle • The opposite and adjacent sides differ depending on the angle of interest. For example, if you are looking at angle X, then the opposite side is a. However, if you are looking at angle Y, then the opposite side is b. Other notes • Never use the right angle when using trigonometric ratios. Only use one of the two acute angles. • Calculators must be in “degree” mode. To check, press the “mode” button and go down to where you see “radian” and “degree”. If not already highlighted, highlight “degree” and press enter. • If your calculator is in “radian” mode, you will not get the correct answers we are looking for here. (They are correct answers but for Geometry we want answers in Degree Mode) Example 1: Write the sine, cosine, and tangent ratios for angles T and U. You Try Write the sine, cosine, and tangent ratios for angles J and K. Finding Side Length • When using trigonometry to solve for a side length, first determine which trig ratio to use based on the given information. • Then, substitute in the information. • Finally, solve as you would solve a proportion. We usually round to the nearest tenth. Example 2: Find the value of x to the nearest tenth. 1) 2) opp sin 40 = hyp x sin 40 = 12 12sin 40 = x opp tan 56 = adj x tan 56 = 12 12 tan 56 = x 7.7 » x 17.8 » x Example 2b: Find the value of x to the nearest tenth. 3) adj hyp x cos 24 = 14 14 cos24 = x cos 24 = 12.8 » x 4) opp hyp x sin 64 = 7 7sin 64 = x sin 64 = 6.3 » x Try these 5) adj cos35 = hyp x cos35 = 19 19 cos35 = x 6) 15.6 » x 7) opp sin19 = hyp x sin19 = 10 10sin19 = x 3.3 » x opp tan 54 = hyp x tan 54 = 10 10 tan 54 = x 13.8 » x What if my variable is in the Denominator? 8) 9) opp hyp 9 sin 23 = x xsin 23 = 9 sin 23 = 9 x= sin 23 x » 23.0 opp tan 75 = adj 21 tan 75 = x xtan 75 = 21 21 tan 75 5.63 » x x= Fun Stuff! 10) adj hyp 9 cos37 = x xcos37 = 9 11) cos37 = 9 x= cos37 x » 11.27 opp hyp 8 sin15 = x xsin15 = 8 sin15 = 8 x= sin15 x » 30.90 Just some more… 12) 13) adj hyp 32 cos 61 = x xcos61 = 32 opp tan 41 = adj 23 tan 41 = x xtan 41 = 23 cos 61 = 23 x= tan 41 x » 26.46 32 x= cos61 x » 66.01 Finding Angle Measures • If given two sides of a right triangle, we can determine the angle measures by using inverse trigonometric ratios. • Start by determining the appropriate ratio to use and substituting in your information. • Then, take the inverse of the ratio. • To do this on the calculator, hit “2nd” and then hit either sin, cos, or tan (depending on which ratio is appropriate given the problem). Ex: Finding Angle Measure Examples: Finding Angles Find the value of x. Round answers to the nearest degree. 1) 2) adj cos x = hyp 10 cos x = 18 æ 10 ö x = cos-1 ç ÷ è 18 ø tan x = x » 56 x » 29 opp adj 11 tan x = 20 æ 11 ö x = tan -1 ç ÷ è 20 ø Examples: Finding Angles 3) opp hyp 12 sin x = 22 æ 12 ö x = sin -1 ç ÷ è 22 ø 4) sin x = cos x = x » 33 x » 68 adj hyp 10 cos x = 27 æ 10 ö x = cos-1 ç ÷ è 27 ø Try these… 5) 6) opp tan x = adj 100 tan x = 41 æ 100 ö x = tan -1 ç ÷ è 41 ø opp sin x = hyp 6.5 sin x = 10 -1 æ 6.5 ö x = sin ç ÷ è 10 ø x » 68 x » 41 Critical Thinking • Describe and create a triangle where: sinÐ = cosÐ Critical Thinking • Given the right triangle ABC where C is the Right Angle, determine if the following statement is valid. Explain why or why not. sin A > tan A