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Math 113 Right Triangle Trigonometry Handout B (length of hypotenuse) - c a - (length of side opposite θ ) θ A C b (length of side adjacent to θ ) Pythagorean’s Theorem: for triangles with a right angle ( side 2 + side 2 = hypotenuse 2 ) a2 + b2 = c2 The definitions of the six trigonometric functions of the acute angle θ are as follows: sin θ = a length of side opposite θ = c length of hypotenuse csc θ = c length of hypotenuse = a length of side opposite θ cos θ = b length of side adjacent to θ = c length of hypotenuse sec θ = c length of hypotenuse = b length of side adjacent to θ tan θ = length of side opposite θ a = b length of side adjacent to θ cot θ = b length of side adjacent to θ = a length of side opposite θ Example: Find the value of each of the six trigonometric functions of θ in the figure below. B a=3 c θ A b=4 C Solution: In order to evaluate all six trigonometric functions, we need to know the length of all sides of the triangle. Since the lengths for sides a and b are given, we can use Pythagorean’s Theorem, c 2 = a 2 + b 2 , to find the length of side c. c 2 = a 2 + b 2 = 32 + 42 = 9 + 16 = 25 c 2 = 25 c = 25 = 5 So, we know that a = 3, b = 4 , and c = 5. We can now use the above definitions of the trigonometric functions to evaluate them for the angle θ . sin θ = a cos θ = b tan θ = a c c b = = = 3 hypotenuse 5 opposite adjacent hypotenuse opposite adjacent = = c csc θ = = 4 5 sec θ = 3 4 cot θ = a c b b a = = = hypotenuse opposite hypotenuse adjacent adjacent opposite = 5 3 = = 5 4 4 3 Two special Right-triangles 1. The “ 45o − 45o − 90o ” right triangle. We can construct a right triangle with a 45o angle. The triangle has two 45o angles. Therefore, the triangle is isosceles – that is, it has two sides of the same length. Assume that each leg of the triangle has length 1. We can find the length of the hypotenuse using Pythagorean’s Theorem. (length of hypotenuse) 2 = 12 + 12 (length of hypotenuse) 2 = 1 + 1 2 (length of hypotenuse) 2 = 2 1 (length of hypotenuse) = 2 45o 1 Now that we know the lengths of the sides of this right triangle, we can find the six trigonometric function values for the angle θ = 45o . sin 45o = cos 45o = tan 45o = opposite hypotenuse adjacent hypotenuse opposite adjacent = = 1 csc 45o = 2 = 1 sec 45o = 2 1 1 cot 45o = hypotenuse opposite hypotenuse adjacent adjacent opposite = 1 1 = = 2 1 2 1 2. The “ 30o − 60o − 90o ” right triangle. There are two other angles that occur frequently in trigonometry, 30o and 60o . We can find the values of the trigonometric functions for these angles using a right triangle. To form this right triangle, draw an equilateral triangle-that is a triangle with all sides the same length. Assume that each side has a length equal to 2. If we draw a line right down the middle of this triangle bisecting the top angle and dividing the base into two equal parts, then we will have a right triangle. See the figure below. We can find the length of the missing side, a, using Pythagorean’s Theorem. 22 = 12 + a 2 30o 2 2 4 = 1 + a2 a 4 − 1 = a2 60o 3 = a2 60o 1 3=a 1 So, our triangle has sides with lengths, 1, 2 , and 3 . Using this right triangle we can find the function values for both 30o and 60o . Fill in the blanks below. opposite sin 30o = hypotenuse 30o 3 2 cos 30o = adjacent hypotenuse = __________ = __________ csc 30o = sec 30o = hypotenuse opposite hypotenuse adjacent = __________ = __________ 60o 1 tan 30o = sin 60o = cos 60o = tan 60o = opposite adjacent opposite hypotenuse adjacent hypotenuse opposite adjacent = __________ cot 30o = adjacent opposite = __________ csc 60o = hypotenuse = __________ sec 60o = hypotenuse cot 60o = adjacent = __________ opposite adjacent opposite = __________ = __________ = __________ = __________ The trigonometric function values for an angle θ depend only on the size of the angle θ , and NOT on the size of the triangle. a=6 a = 4.5 a=3 θ θ b=2 tan θ = a 3 = b 2 a = 1.5 b=4 tan θ = a 3 = b 2 θ θ b =1 tan θ = a 3 = b 2 b=3 tan θ = a 3 = b 2 Notice that all of these right triangles have the same angle, θ . Even though the triangles are different sizes, they are “similar”. This means that the triangles have the same shape and the lengths of the corresponding sides are in the same ratio. Because an acute angle in a right triangle always gives the same ratio of opposite to adjacent sides, the trigonometric functions’ values evaluated for the angle θ will be the same for all of these triangles. Since we know that the size of the triangle is not important, it is helpful to look at right triangles in which the length of the hypotenuse is equal to 1. Here are our 2 special triangles adjusted so that the length of the hypotenuse is 1. The Basic “ 30o − 60o − 90o ” Right Triangle The Basic “ 45o − 45o − 90o ” Right Triangle 30o 45o 3 2 1 60o 1 1 2 = 2 2 45o 1 2 1 2 = 2 2 Facts: For any “ 30o − 60o − 90o ” the length of the shortest leg is always For any “ 30o − 60o − 90o ” the length of the longest leg is always For any “ 45o − 45o − 90o ” the length of the hypotenuse is always 1 times the length of the hypotenuse. 2 3 times the length of the shortest leg. 2 times the length of a leg.