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13.1 Right Triangle Trigonometry ©2002 by R. Villar All Rights Reserved Right Triangle Trigonometry Let’s consider a right triangle, one of whose acute angles is The three sides of the triangle are the hypotenuse, the side opposite , and the side adjacent to . Ratios of a right triangle’s three sides are used to define six trigonometric functions: hypotenuse sine, cosine, tangent, cosecant, secant, and cotangent opposite Right Triangle Definition of Trig Functions Let be an acute angle of a right triangle. sin = opp hyp cos = adj hyp tan = opp adj csc = hyp opp sec = hyp adj cot = adj opp adjacent Example: Evaluate the six trigonometric functions of the angle shown. Use the Pythagorean Theorem to find the length of the adjacent side… a2 + 122 = 132 a2 = 25 a=5 adj = 5 opp = 12 hyp = 13 12 13 sin = opp hyp = 12 13 cos = adj = 5 13 hyp tan = opp = 5 12 adj csc = hyp opp = 13 12 sec = hyp = adj cot = adj = opp 13 5 12 5 Example: Given cos = 4/5 , find cot . Since cos = adj = 4 , the triangle looks like this… hyp 5 5 Use the Pythagorean Theorem to find the length of the opposite side… a2 + 42 = 52 a2 = 9 a=3 adj = 4 opp = 3 hyp = 5 cot = adj = opp 4 3 4 The angles 30º , 45º , and 60º occur frequently in trigonometry. Use the triangle to find the trig. functions when = 30º adj = 3 opp = 1 hyp = 2 30º 2 3 1 sin 30º = opp = hyp 1 2 cos 30º = adj 3 2 hyp tan 30º = opp 1 3 3 3 adj csc 30º = hyp = opp 2 1 2 3 sec 30º = hyp 2 3 3 adj cot 30º = Remember, adj 3 the radical in the denominator 1 opp must be simplified. Notice that the same triangle can be used when the acute angle is 60º : Use the triangle to find the trig. functions when = 60º adj = 1 opp = 3 hyp = 2 2 3 60º 1 sin 60º = opp 3 2 hyp cos 60º = adj = hyp csc 60º = hyp 2 2 3 3 3 opp sec 60º = hyp = adj tan 60º = opp 3 1 adj 1 2 2 1 1 cot 60º = adj 3 opp 3 3 What if the acute angle is 45º ? Use the triangle to find the trig. functions when = 45º adj = 1 opp = 1 hyp = 2 2 1 45º 1 sin 45º = opp 1 2 2 2 hyp 2 1 cos 45º = adj 2 hyp 2 tan 45º = opp 1 1 adj 1 2 csc 45º = hyp 1 opp sec 45º = hyp 2 1 adj cot 45º = adj 1 1 opp 1 You will need to remember how to re-create these triangles.