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13.1 Use Trigonometry with Right Triangles Goal Use trigonometric functions to find lengths. Your Notes VOCABULARY Sine, cosine, tangent, cosecant, secant, cotangent The six trigonometric functions that consist of ratios of a right triangle’s side lengths RIGHT TRIANGLE DEFINITIONS OF TRIGONOMETRIC FUNCTIONS Let be an acute angle of a right triangle. The six trigonometric functions of are defined as follows: hypotenuse csc = opposite opposite sin = hypotenuse cos = adjacent hypotenus sec = opposite adjacent tan = hypotenuse adjacent adjacen cot = opposit The abbreviations opp, adj, and hyp are often used to represent the side lengths of the right triangle. Note that the ratios in the second column are reciprocals of the ratios in the first column: csc = 1 sin sec = 1 cos cot = 1 tan Your Notes Example 1 Evaluate trigonometric functions Evaluate the six trigonometric functions of the angle 0. From the Pythagorean theorem, the length of the hypotenuse is 2 289_17_. 82 15 = opp hy 8 17 Sin = csc hy opp cos = adj hy tan = opp adj sec = hyp 15 17 adj cot = adj 8 15 opp 1 8 17 15 15 8 TRIGONOMETRIC VALUES FOR SPECIAL ANGLES The table below gives the values of the six trigonometric functions for the angles 30°, 45°, and 60°. You can obtain these values from the triangles shown. 0 30 sin 45 60 cos tan 3 2 3 3 1 2 2 2 1 2 2 1 2 3 2 csc 30 2 sec 3 cot 2 3 3 45 2 60 2 3 3 2 2 3 1 3 3 Your Notes Example 2 Use a calculator to solve a right triangle Solve ABC. Solution A and B are complemetary angels, so A = 90 _56_ = _34_ opp sec hyp 56 = ad b c = sec 56 13 13 tan 56 = ad tan 56 = __13(tan 56)__ = b _13_ _19.27_ b 1 co 56 c _23.25_ c So, A = _34_, b _19.27_, and c _23.25_. Checkpoint Complete the following exercises. 1. Evaluate the six trigonometric functions of the angle 0. 3 13 sin 0 = csc = 13 cos 0 = 2 13 13 13 3 sec = 13 2 tan = 3 cot = 2 2 3 2. Solve ABC. B = 53, a 13.24, and b 17.57 Your Notes Example 3 Use an angle of elevation Building Height You are measuring the height of your school building. You stand 25 feet from the base of the school. The angle of elevation from a point on the ground to the top of the school is 62°. Estimate the height of the school to the nearest foot. Solution 1. Draw a diagram that represents the situation. 2. Write and solve an equation to find the height h. tan _62_ = h 25 _25_(tan 62) = h _47_ h The height of the school is about _47_ feet. Checkpoint Complete the following exercise. 3. A kite makes an angle of 59° with the ground. If the string on the kite is 40 feet, how far above the ground is the kite itself? Round to the nearest foot. About 34 ft Homework ______________________________________________________________________________________ ______________________________________________________________________________________