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. hypotenuse opposite adjacent Ratios of a right triangle’s three sides are used to define six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent 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

Example: Evaluate the six trigonometric functions of the angle shown. Use the Pythagorean Theorem to find the length of the adjacent side… a = 13 2 a 2 = 25 a = 5 adj = 5opp = 12hyp = 13 = sin =opp hyp cos =adj hyp = 5 13 tan =opp adj = 5 12 csc =hyp opp = sec =hyp adj = 13 5 cot =adj opp = 12 5

Example: Given cos = 4/5, find cot. Since cos = adj = 4, the triangle looks like this… hyp 5 5 cot = adj opp 4 Use the Pythagorean Theorem to find the length of the opposite side… a = 5 2 a2 = 9 a2 = 9 a = 3 adj = 4opp = 3hyp = 5 = 4 3

The angles 30º, 45º, and 60º occur frequently in trigonometry. Use the triangle to find the trig. functions when = 30º = sin 30º = opp hyp cos 30º = adj hyp tan 30º = opp adj csc 30º = hyp opp = 2 1 sec 30º = hyp adj cot 30º = adj opp adj = opp = 1hyp = 2 Remember, the radical in the denominator must be simplified. 30º

Notice that the same triangle can be used when the acute angle is 60º : Use the triangle to find the trig. functions when = 60º = sin 60º = opp hyp cos 60º = adj hyp tan 60º = opp adj csc 60º = hyp opp = 2 1 sec 60º = hyp adj cot 60º = adj opp adj = 1opp = hyp = 2 60º

What if the acute angle is 45º ? Use the triangle to find the trig. functions when = 45º 1 sin 45º = opp hyp cos 45º = adj hyp tan 45º = opp adj csc 45º = hyp opp sec 45º = hyp adj cot 45º = adj opp adj = 1opp = 1hyp = 45º 1 You will need to remember how to re-create these triangles.