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Chapter 3 Trigonometric Functions of Angles Section 3.2 Trigonometry of Right Triangles Hypotenuse, Adjacent and Opposite sides of a Triangle In a right triangle (a triangle with a right angle) the side that does not make up the right angle is called the hypotenuse. For an angle that is not the right angle the other two sides are names in relation to it. The opposite side is a side that makes up the right angle that is across from . The adjacent side is the side that makes up the right angle that also forms the angle . hypotenuse opposite side adjacent side hypotenuse adjacent side opposite side The Trigonometric Ratios For any right triangle if we pick a certain angle we can form six different ratios of the lengths of the sides. They are the sine, cosine, tangent, cotangent, secant and cosecant (abbreviated sin, cos, tan, cot, sec, csc respectively). hypotenuse opposite side adjacent side sin opposite side hypotenuse tan opposite side adjacent side sec hypotenuse adjacent side cos adjacent side hypotenuse cot adjacent side opposite side csc hypotenuse opposite side To find the trigonometric ratios when the lengths of the sides of a right triangle are know is a matter of identifying which lengths represent the hypotenuse, adjacent and opposite sides. In the triangle below the sides are of length 5, 12 and 13. We want to find the six trigonometric ratios for each of its angles and . sin 5 13 cos 12 13 tan 5 12 cot 12 5 sec 13 12 csc 13 5 13 5 12 Notice the following are equal: sin cos sin cos tan cot tan cot sec csc sec csc sin 12 13 cos 5 13 tan 12 5 cot 5 12 sec 13 5 csc 13 12 The angles and are called complementary angles (i.e. they sum up to 90). The “co” in cosine, cotangent and cosecant stands for complementary. They refer to the fact that for complementary angles the complementary trigonometric ratios will be equal. Pythagorean Theorem and Trigonometric Ratios c The Pythagorean Theorem relates the sides of a right triangle so that if you know any two sides of the triangle you can find the remaining one. This is particularly useful in trig since two sides will then determine all six trigonometric ratios. b a a2 b2 c2 Determine the six trigonometric ratios for the right triangle pictured below. sin cos tan 13 7 6 7 cot 13 6 6 13 sec 7 6 csc 7 13 x 7 sin 6 7 cos 13 7 6 13 tan 6 cot First we need to determine the length of the remaining side which we will call x and apply the Pythagorean Theorem. x 2 62 7 2 x 2 36 49 x 2 13 x 13 sec 13 6 7 13 csc 7 6 Finding Other Trigonometric Ratios by Knowing One If one of the trigonometric ratios is known it is possible to find the other five trigonometric ratios by constructing a right triangle with an angle and sides corresponding to the ration given. For example, if we know that the sin = ¾ find the other trigonometric ratios. 1. Make a right triangle and label one angle . 2. Make the hypotenuse length 4 and the opposite side length 3. 3. Find the length of the remaining side. 4. Find the other trigonometric ratios. 4 3 x= 7 sin 3 4 cos 7 4 3 7 tan x 2 32 4 2 x 2 9 16 x2 7 x 7 cot sec 7 3 4 7 csc 4 3 Similar Triangles and Trigonometric Ratios Triangles that have the exact same angles measures but whose sides can be of different length are called similar triangles. Similar triangles have sides that are proportional. That is to say the sides are just a multiple of each other. Consider the example above where the sides of one triangle are just three times longer than the side of the other triangle. 5 3 15 4 9 sin 9 3 15 5 cos 12 4 15 5 tan 9 3 12 4 cot 12 4 9 3 sec 15 5 12 4 csc 15 5 9 3 12 Similar Triangles have equal trigonometric ratios! Many of the ideas in trigonometry are based on this concept. Trigonometric Ratios of Special Angles 45-45-90 Triangles 30-60-90 Triangles If you consider a square where each side is of length 1 then the diagonal is of length 2 . If you consider an equilateral triangle where each side is of length 1 then the perpendicular to the other side is of length 3 . 1 1 x 2 45 1 x 2 2 11 x2 2 x 2 12 12 2 2 1 2 x2 45 1 x 1 2 2 2 cos 45 tan 45 1 2 2 2 1 1 1 cot 45 1 1 1 sec 45 2 1 csc 45 2 1 x2 3 2 x 60 1 2 2x sin 45 x 2 14 1 30 3 4 3 2 sin 60 3 2 sin 30 1 2 cos 60 1 2 cos 30 3 2 tan 60 3 tan 30 1 3 3 3 3 cot 60 1 3 3 3 cot 30 2 sec 60 2 sec 30 2 csc 60 2 2 3 3 3 csc 30 2 2 3 3 3 2 Finding the Length of Sides of Right Triangles If you know the length of one of the sides and the measure of one of the angles of a right triangle you can find the length of the other sides by using trigonometric ratios. 8 x z 17 37 y 68 x Find the values for x and y. Find the values for x and z. x sin 37 8 x 8 sin 37 x = 4.81452 x cot 68 17 x 17 cot 68 y cos 37 8 y = 6.38908 y 8 cos 37 z csc 68 17 z 17 csc 68 x = 6.86845 z = 18.3351