Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Definition III: Circular Functions Trigonometry MATH 103 S. Rook Overview • Section 3.3 in the textbook: – The six trigonometric functions and the Unit Circle – Domain of the six trigonometric functions 2 The Six Trigonometric Functions and the Unit Circle Unit Circle • Recall that the Unit Circle is the special circle with a radius of 1 and equation of x2 + y2 = 1 4 Revisiting Definition I and Definition II • Consider a point (x, y) on the unit circle • By Definition I: cos x x x r 1 sin y y y r 1 • By Definition II: adj x cos x hyp 1 opp y sin y hyp 1 5 Circular Functions • Therefore, (x, y) can be written as (cos θ, sin θ) • Now consider a point (x, y) on the circumference of the unit circle where t is the length of the arc from (1, 0) to (x, y) s t • Then r 1 t – The central angle is equivalent to the length of the arc it cuts on the Unit Circle 6 Circular Functions (Continued) • All points encountered on the unit circle can then be written as (cos t, sin t) where t is the distance traveled from (1, 0) t can be positive (counterclockwise) or negative (clockwise) • The six trigonometric functions with respect to the Unit Circle are: cos t x sin t y y tan t , x 0 x x cot t , y 0 y 1 csc t , y 0 y 1 sec t , x 0 x 7 Circular Functions (Continued) • We call these the circular functions – The radian measure of θ is the same as the arc length from (1, 0) to a point P on the terminal side of θ on the circumference of the unit circle 8 Circular Functions (Example) Ex 1: Use the Unit Circle to find the six trigonometric functions of: a) 7 4 b) 4 3 9 Circular Functions (Example) Ex 2: Use the Unit circle to find all values of t, 0 < t < 2π where a) 1 cos t 2 b) sin t 2 2 c) tan t 1 10 Circular Functions (Example) Ex 3: If t is the positive distance from (1, 0) to point P along the circumference of the unit circle, sketch t on the circumference of the unit circle and then find the value of: a) P = (-0.9422, 0.3350); find i) cos t, ii) csc t, and iii) cot t b) P = (0.5231, -0.8523); find i) sin t, ii) sec t, and iii) tan t 11 Review of Functions • Recall that a function can be thought of as a machine which takes an input (or an argument) value and operates on it to produce an output value – Ex. y = cos x: input x is placed into the cosine function to produce output y 12 Review of Functions (Example) Ex 4: Identify i) the function, ii) the argument of the function, and iii) the function value: a) b) sin 13 6 tan 4 13 Domain of the Six Trigonometric Functions Domain of cosine and sine • Recall that the domain of a function is the set of allowable input values (usually x) • For cos t and sin t, there are no domain restrictions – cosine and sine are defined for every value of t along the circumference of the unit circle – Written in interval notation as , 15 Domain of secant and tangent • For sec t and tan t, 1 sec t cos t and sin t tan t cos t – Undefined when cos t = 0 • Referring to the Unit Circle, cos t 0 when t 2 or • Starting at 2 , cos t = 0 every additional π radians 3 2 (multiple of π) • Therefore, sec t and tan t are undefined whenever t 2 k where k is an integer and their domain is: t | t R, t k where k is an integer 2 16 Domain of the cosecant and tangent • For csc t and cot t, 1 csc t sin t and cos t cot t sin t – Undefined when sin t = 0 • Referring to the Unit Circle, sin t 0 when t 0 or • Starting at 0, sin t = 0 every additional π radians (multiple of π) • Therefore, csc t and cot t are undefined whenever t k where k is an integer and their domain is: t | t R, t k where k is an integer 17 Domain of the Trigonometric Functions (Example) Ex 5: Answer the following: a) What is the domain for the secant and the tangent? What is the value of tan 7 2 b) What is the domain for the cosecant and the cotangent? What is the value of csc( 6 ) 18 Summary • After studying these slides, you should be able to: – Define the six trigonometric functions in terms of circular functions – Identify the domain of the six trigonometric functions • Additional Practice – See the list of suggested problems for 3.3 • Next lesson – Arc Length and Area of a Sector (Section 3.4) 19