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Chief SOHCAHTOA Objective: I will be able to: Define the three basic trig ratios and their reciprocal functions. Given two sides of a right triangle, find the third side and write all 6 trig ratios in simplest form. Given two trig ratios for a triangle, find the third side and write all 6 trig ratios in simplest form. Given one trig ratio for a triangle, find the third side and write all 6 trig ratios in simplest form. FINDING A MISSING SIDE OF A TRIANGLE--Review When working with right triangles, it is often necessary to use the Pythagorean Theorem. Remember, if you are given ANY two sides, then the Pythagorean Theorem will allow you to find the missing side. 2 2 a2+b2=c2 ----- In other words leg1 leg 2 hyp 2 . THREE BASIC TRIG RATIOS--Definitions The three basic trig ratios can be used for ANY right triangle. You may use Chief SohCahToa’s name to help you remember them. In trigonometry, we frequently use the Greek letters theta, , and alpha, , to represent angles. The three ratios are sine, cosine, and tangent. We use their abbreviations sin, cos, and tan. However, these abbreviations are ALWAYS followed by an angle name or measure. Never just write sin, cos, or tan. opp sin Hypotenuse hyp Opp—This is the adj side opposite to the cos hyp angle . opp tan adj Adj--- This is the side ADJACENT to the angle RECIPROCAL TRIG RATIOS In addition to the three basic trig ratios, there are three RECIPROCAL ratios. These reciprocal ratios are cosecant, secant, and cotangent. We use their abbreviations csc, sec, and cot. However, these abbreviations are ALWAYS followed by an angle name or measure. The reciprocal of sine is cosecant. The reciprocal of cosine is secant. The reciprocal of tangent is cotangent. opp hyp sin csc hyp opp adj hyp cos sec hyp adj opp adj tan cot adj opp NOTE: The two words that begin with “co” never together! Given 2 sides of a right triangle, find the 3rd side & write all 6 trig ratios in simplest form. Given: 2 1 Step 1: Label the triangle. 2 hyp 1 opp Step 2: Find the missing side by using Pythagorean Theorem. adj2 + opp2 = hyp2 1 adj2 + 12= 22 opp 2 adj = 3 adj adj = 3 adj = Step 3: Write the three basic ratios. 1 sin 2 3 cos 2 1 3 tan which equals tan 3 3 2 hyp 3 Step 4 Write the three reciprocal ratios. 2 csc 1 2 2 3 sec which equals sec 3 3 cot 3 1 Given two trig ratios for a triangle, find the third side and write all 6 trig ratios in simplest form. 2 5 cos Given sin 3 3 Step 2: Check to see if it labeled correctly by Step 1: Label the triangle. using Pythagorean Theorem. hyp=3 adj2 + opp2 = hyp2 opp=2 22 + 5 2= 32 4+5= 9 adj= 5 Step 4 Write the three reciprocal ratios. 3 csc 2 3 3 5 sec which equals sec 5 5 Step 3: Write the three basic ratios. 2 sin 3 5 cos 3 tan 2 5 which equals tan 2 5 5 cot 5 2 Given one trig ratio for a triangle, find the third side and write all 6 trig ratios in simplest form. hyp Given: sec 3 Remember sec adj Step 2: Find the missing side by using Pythagorean Theorem. Step 1: Label the triangle. adj2 + opp2 = hyp2 hyp=3 hyp=3 opp 12 + opp2= 32 adj=1 Step 3: Write the three basic ratios. 2 2 sin 3 1 cos 3 2 2 tan 1 opp2= 8 opp = 8 =2 2 opp = 2 2 Step 4 Write the three reciprocal ratios. 3 3 2 csc which equals 4 2 2 3 sec 1 1 2 cot which equals 4 2 2 adj=1