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Precalculus Trigonometric Functions The Unit Circle 2015 Precalculus WU 10/14 Find one positive and one negative angle co-terminal with the given angle. 5 12 Objectives: • Find trig function values for special angles using the unit circle. • Evaluate Trig functions using the unit circle. • Use domain and period to evaluate trig functions. • Solve application problems using the unit circle. The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are: hyp the side opposite the acute angle , opp the side adjacent to the acute angle , θ and the hypotenuse of the right triangle. adj The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. opp sin = cos = adj tan = opp hyp hyp adj csc = hyp opp sec = hyp adj Copyright © by Houghton Mifflin Company, Inc. All rights reserved. cot = adj opp 4 Trigonometric Functions Let be an angle in standard position with (x, y), a point on the terminal side of and r = x 2 y 2 0. sin y r cos x r y y , x0 x cot x , y 0 y sec r , x 0 x csc r , y 0 y tan (x, y) x 5 Example: Determine the exact values of the six trigonometric functions of the angle . y r x 2 y 2 (3) 2 62 (3, 6) 9 36 45 x y 6 2 5 r 5 45 cos x 3 5 r 5 45 y tan 6 2 x 3 sin csc r 45 5 y 6 2 sec r 45 5 x 3 cot x 3 0.5 y 6 6 So, we know that trigonometric function values are side length relationships of right triangles. We can easily evaluate the exact values of trigonometric functions for special angles. Geometry of the 30-60-90 triangle Consider an equilateral triangle with each side of length 2. 30○ 30○ The three sides are equal, so the angles are equal; each is 60. 2 The perpendicular bisector of the base bisects the opposite angle. 60○ 2 3 1 60○ 2 1 Use the Pythagorean Theorem to find the length of the altitude, 3 . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Special right triangle relationships 2 3 2 1 45 60 1 1 Now, let’s apply it to the unit circle… What does “unit circle” really mean? It’s a circle with a radius of 1 unit. What is the equation of the “unit circle”? x y 1 2 2 0,1 2 0, 0 1,0 2 , 360 , 180 -1,0 0, -1 3 2 Let’s begin with an easy family… 4 What are the coordinates? 2 4 1 , 180 2 2 2 2 45 2 2 3 2 Now, reflect the triangle to the second quadrant… 0, 0 2 , 360 , 2 2 What are the coordinates? - 2 2 , 2 2 2 3 4 4 1 2 2 1 - 2 2 2 2 2 2 2 45 , 180 2 3 2 Now, reflect the triangle to the third quadrant… 0, 0 2 , 360 , 2 2 2 - 2 , 2 2 2 3 4 4 1 2 2 - - 2 2 , - 2 2 1 2 2 2 2 2 5 4 2 2 2 45 , 180 What are the coordinates? 3 2 Now, reflect the triangle to the fourth quadrant… 0, 0 2 , 360 , 2 2 2 - 2 , 2 2 2 3 4 4 1 2 2 - - 2 , - 2 2 1 7 3 2 , 2 2 2 0, 0 2 , 360 2 2 5 4 2 2 2 2 2 45 , 180 2 4 What are the coordinates? 2 2 , - 2 2 Complete the family… 6 . 6 1 30 3 2 , 1 2 1 2 3 2 Now, reflect the triangle to the second quadrant. 3 1 , 2 2 6 5 6 1 2 - 3 2 1 1 30 2 3 2 , 1 2 3 2 Now, reflect the triangle to the third quadrant. 3 1 , 2 2 5 6 1 2 - What are the coordinates? - 3 2 , - 1 2 6 7 3 2 1 1 30 2 3 2 , 3 2 6 Now, reflect the triangle to the fourth quadrant. 1 2 3 1 , 2 2 5 6 1 2 - - 3 2 , - 1 2 6 7 6 3 2 1 1 30 2 3 2 , 1 2 3 2 11 3 2 , - 1 2 6 What are the coordinates? Let’s look at another “family” 3 2 3 2 , 3 2 3 1 2 , 180 1 60 1 2 3 2 Now, reflect the triangle to the second quadrant 0, 0 2 , 360 What are the coordinates? - 1 2 , 3 2 2 2 3 3 1 3 2 - 1 60 1 2 2 , 3 2 3 1 2 , 180 1 2 3 2 Now, reflect the triangle to the third quadrant 0, 0 2 , 360 - 1 2 3 , 2 2 3 1 2 - 1 60 1 2 2 What are the coordinates? - 2 , - 2 2 , 3 2 4 3 3 1 2 , 180 3 1 3 3 1 2 3 2 Now, reflect the triangle to the fourth quadrant 0, 0 2 , 360 - 1 2 3 , 2 2 3 1 2 - - 2 , - 2 2 , 3 2 4 3 3 1 2 , 180 3 1 3 3 1 2 1 0, 0 2 , 360 60 1 2 2 3 2 5 3 What are the coordinates? 1 2 , - 3 2 Ordered pairs of special angles around the Unit Circle 1 3 2, 2 Since r = 1… cos x 1 3,1 2 2 (–1, 0) 3 1 2 , 2 2 2 , x, y cos ,sin 90° 2 3 3 120° 5 4 135° 6 150° 2 2 2 , 2 y sin 1 y 180° (0, 1) 2 60° 3 1 3 2, 2 2 2 2 , 2 4 45° 30° 6 3 1 2 ,2 0° 0 x 360° 2 (1, 0) 330° 11 315° 3 1 7 210° 6 6 2 , 2 225° 7 5 240° 300° 2 4 4 5 4 2 , 2 2 2 2 3 270° 32 3 1 3 , 2 2 (0, –1) 1 3 , 2 2 24 • Important point: Since r = 1… sin y 1 cos x 1 x, y cos ,sin Because ordered pairs around the unit circle (x, y) represent 2 2 x y 1, sine and cosine, and the equation of the circle is We have the following identity: cos 2 sin 2 1 What if the radius is not 1? 6 1 30 30 Trigonometric values are functions of the angle – ratios of sides of similar triangles remain the same. So it always holds that cos 2 sin 2 1. Trigonometric Values of Special Angles 1 3 2, 2 (–1, 0) 3 1 2 , 2 2 2 , 90° 2 3 3 120° 5 4 135° 6 150° 2 2 2 , 2 3,1 2 2 y 180° (0, 1) 2 60° 3 1 3 2, 2 2 2 2 , 2 4 45° 30° 6 3 1 2 ,2 0° 0 x 360° 2 (1, 0) 330° 11 315° 3 1 7 210° 6 6 2 , 2 225° 7 5 240° 300° 2 4 4 5 4 2 , 2 2 2 2 3 270° 32 3 1 3 , 2 2 (0, –1) 1 3 , 2 2 27 Domain The domain of the sine and cosine function is the set of all real numbers. (0, 1) y Unit Circle (–1, 0) (1, 0) x 1 y 1 (0, –1) 1 x 1 Range The point (x, y) is on the unit circle, therefore the range of the sine and cosine function is between – 1 and 1 inclusive. 28 A function f is periodic if there is a positive real number c such that f (t + c) = f (t) for all t in the domain of f. The least number c for which f is periodic is called the period of f. y Unit Circle f (t) sin t Periodic Function x t = 0, 2, … Period 29 Example: Evaluate sin 5 using its period. 5 - 2 - 2 = sin 5 = sin = 0 f (t) sin t y (–1, 0) x Adding 2 to each value of t in the interval [0, 2] completes another revolution around the unit circle. sin(t 2 n) sin t cos(t 2 n) cos t 30 You Try: 9 a) Evaluate sin 4 b) Evaluate cos 10 3 Can you? Evaluate each of the following. Exact values only please. 5 tan 6 3 sec 4 Even and Odd Trig Functions Remember: if f(-t) = f(t) the function is even if f(-t) = - f(t) the function is odd The cosine and secant functions are EVEN. cos(-t)=cos t sec(-t)=sec t (0,1) y (–1, 0) (0,–1) The sine, cosecant, tangent, and cotangent functions are ODD. sin(-t)= -sin t csc(-t)= -csc t tan(-t)= -tan t cot(-t)= -cot t (1, 0) Example: Evaluate the six trigonometric functions at = . sin y 0 cos x 1 y 0 tan 0 x 1 csc 1 1 is undefined. y 0 sec 1 1 1 x 1 cot x 1 is undefined. y 0 (0, 1) y (–1, 0) x (1, 0) (0, –1) 34 Application: A ladder 20 feet long leans against the side of a house. The angle of elevation of the ladder is 60 degrees. Find the height from the top of the ladder to the ground. Application: An airplane flies at an altitude of 6 miles toward a point directly over an observer. If the angle of elevation from the observer to the plane is 45 degrees, find the horizontal distance between the observer and the plane. . Homework 4.2 pg. 264 1-51 odd Trig Races 4 cos 3 1 2 3 3 tan210 11 csc 4 2 csc7 undef. 22 sin 3 cos 3 3 2 1 2 7 sin 3 3 sin 2 4 sec 3 3 2 3 csc 4 1 7 tan 3 2 8 cot 3 2 3 3 3 HWQ 10/15 Evaluate each of the following. Exact values only please. 5 tan 6 3 sec 4 csc7 22 sin 3