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Trigonometry A Pre-calculus Preview Ch 13 (1,2,4) Notes Vocabulary: Trigonometric function, sine, cosine, tangent, cosecant, secant, cotangent, standard position, initial side, terminal side, angle of rotation, coterminal angles, reference angles, inverse sine function, inverse cosine function, inverse tangent function A Trigonometric function is a function whose rule is given by a trigonometric ratio. A Trigonometric ratio compares the lengths of two sides of a right triangle. The Greek letter theta is often used to represent the acute angles of the right triangle. The values of trigonometric ratios depend on . 13.1 Notes (Right Angle Trigonometry) Trigonometric Functions and Reciprocal Functions (use the Pythagorean Theorem to find missing side of right triangle a 2 b 2 c 2 remember c is the hypotenuse which is across from the right angle) sine = opposite hypotenuse cosine = cosecant = hypotenuse opposite secant = hypotenuse adjacent adjacent hypotenuse Example: 5 3 sin = 3 5 csc = 3 5 cos = 4 5 sec = 4 5 tan = 3 4 tangent = opposite adjacent cotangent = 4 adjacent opposite cot = 3 4 Special Right Triangles You will frequently need to determine the value of trigonometric ratios for 30, 60, and 45 angles. Recall from geometry that the ratio of side lengths for a 30 60 90 is 1: 3 : 2 and for a 45 45 90 the ratio is 1:1: 2 . sin 30 = 1 2 sin 60 = 3 2 1 2 60 30 45 cos 30 = 3 2 cos 60 = tan 30 = 1 3 3 3 tan 60 = 45 sin 45 = 1 2 2 2 tan 45 = 90 cos 45 = 1 2 2 2 90 * The angle of elevation and depression are alternate interior angles therefore have the same measure. 3 3 1 1 1 1 13.2 Notes (Angles of Rotation) An angle is in standard position when its vertex is at the origin and one ray is on the positive x-axis. The initial side of the angle is the ray on the x-axis and the other ray is called the terminal side. An angle of rotation is formed by rotating the terminal side and keeping the initial side in place. Coterminal angles are angles in standard position with the same terminal side. The reference angle is the positive acute angle formed by the terminal side of and the x-axis. Coterminal Angles Reference Angle To determine the value of the trigonometric functions for an angle in standard position you begin by selecting a point P with coordinates (x,y) on the terminal side of the angle. The distance r from point P to the origin is given by x2 y2 . sine = O y H r cosine = A x H r tangent = O y A x The Must Know Chart!!!! angle cos (x) sin (y) tan (y/x) 30 or 6 3 2 1 2 3 3 45 or 4 2 2 2 2 1 60 or 3 3 2 3 1 2 13.4 Notes (Inverses of Trigonometric Functions) To find the measurement of angles given the value of the trigonometric function you must use an inverse trigonometric relation. * Inverse trig functions are also called arcsine, arccosine, and arctangent. Function (to find value) Inverse Relation (to find angle) sin a sin 1 a cos a cos 1 a tan a tan 1 a *Calculators must be set to degrees Example: A group of hikers wants to walk from a lake to an unusual rock formation. The formation is 1 mile east and 0.75 mile north of the lake. To the nearest degree, in what direction should the hikers head from the lake to reach the rock formation? Rock Step 1: Draw a diagram. The hikers’ direction should be based on , the measure of an acute angle of a right triangle. 0.75 mi Step 2: Find the value of . tan θ Lake opp. .75 0.75 adj. 1 1 mi Tan 1 0.75 N 37E The hikers should head north 37 east. Inverses of trigonometric functions are not functions themselves. In order to define inverses as functions you must restrict the domains. When this restriction is used it is noted by using a capital letter. Inverse Trigonometric Functions Symbol Domain Range Sin 1a 1,1 90,90 or 2 , 2 Cos 1a 1,1 0,180 0, Tan 1a , 90,90 or or Quadrant Restriction 2 , 2 Homework: *don’t forget to rationalize the denominators and simplify radicals 13.1 Day 1: page 933-34 (13-18, 21-23) 13.1 Day 2: page 933-34 (8, 9, 19, 20, 25, 26, 27, 30-32) 13.2 Day 1: page 939 (26-41) 13.2 Day 2: page 939 (42-49) 13.4 Day 1: page 953-54 (11, 25, 30, 31, 33, 40)