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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 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
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 . 5 12 13 sin cos tan cot sec csc sin cos tan cot sec csc Notice the following are equal: 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 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. a b c Determine the six trigonometric ratios for the right triangle pictured below. 6 7 First we need to determine the length of the remaining side which we will call x and apply the Pythagorean Theorem. x sin cos tan cot sec csc sin cos tan cot sec csc
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 = sin cos tan cot sec csc
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. 3 9 4 5 15 12 sin cos tan cot sec csc 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 If you consider a square where each side is of length 1 then the diagonal is of length. 1 1 45 sin 45 cos 45 tan 45 cot 45 sec 45 csc 45 30-60-90 Triangles If you consider an equilateral triangle where each side is of length 1 then the perpendicular to the other side is of length. 1 60 sin 60 cos 60 tan 60 cot 60 sec 60 csc 60 sin 30 cos 30 tan 30 cot 30 sec 30 csc 30 30
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. 37 8 x y Find the values for x and y. 17 68 x z Find the values for x and z. x = 4.81452 x = 6.86845 y = 6.38908 z = 18.3351
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