Chapter 4 Pre-Calculus OHHS

4.3 The Circular Functions Solve Trig Functions of Any Angle Solve Trig Functions of Real Numbers Understand Periodic Functions Analyze the 16-point Unit Circle 4-3

Anatomy of an Angle Terminal Side Vertex Initial Side 4-3

Angle Rotation Negative Angle Positive Angle Counter Clockwise 4-3

Standard Position y Terminal Side Initial Side x Vertex at (0,0) 4-3

Quadrantal Angles Terminal side of a standard position angle is on an axis. 4-3

Coterminal Angles 45º+360ºn where n is an integer. Angles with the same initial and terminal sides, but with different rotations. All of these are coterminal angles. 45º 405º -315º 765º -675º -1035º 1485º -1395º How did I find all these angles? 45º+360ºn where n is an integer. 4-3

Example Find a positive angle and a negative angle that are coterminal with (a) 30 30+360 = 390 (b) 30-360=-330 4-3

Now You Try P. 381, #1 4-3

1st Quadrant Trig sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = r y P(x,y) r y θ x csc θ = sec θ = cot θ = 4-3

Example Let θ be the acute angle in standard position whose terminal side contains the point (5, 3). Find the six trigonometric functions of θ. P(5,3) 3 θ 5 4-3

Now You Try P. 381, #5 4-3

Example Let θ be the acute angle in standard position whose terminal side contains the point (-5, 3). Find the six trigonometric functions of θ. P(-5,3) 3 θ -5 4-3

Now You Try P. 381, #11 4-3

Reference Angle The angle formed between the terminal side and the nearest part of the x-axis. 4-3

Example Find the exact values of the six trigonometric functions of 315 Reference Angle 315 45 1 45 -1 4-3

Now You Try P. 381, #25 4-3

Example Find the coordinates on the unit circle where θ = 210º x y 1 sin 210º = cos 210º = tan 210º = sec 210º = csc 210º = cot 210º = θ = 210º x 30º y 1 4-3

Now You Try P. 381, #29 4-3

Trig Functions of Quadrantal Angles cot(180) = undefined 180 (-1, 0) sin(180) = 4-3

Now You Try P. 381, #41 4-3

Using One Trig Ratio to Find the Others If sin θ = and tan θ < 0, find cos θ. In which quadrant is this true? Q2 7 3 θ 4-3

Using One Trig Ratio to Find the Others If sec θ = 3 and sin θ > 0, find tan θ. In which quadrant is this true? Q1 3 1 4-3

Now You Try P. 381, #43 4-3

The Unit Circle 2 A circle with radius = 1 What is its circumference? 4-3

The Unit Circle Wrapping Function Circumference 2  or 2  2 4-3

Using the Unit Circle to Find Trig Ratios The terminal side of any angle t intersects the unit circle at (cos t, sin t) 4-3

Trigonometric Functions on the Unit Circle 4-3

Unit Circle Example Find tan   (-1, 0) 4-3

Periodic Functions f(t + c) = f(t) A function is periodic if there is a positive number c such that f(t + c) = f(t) for all values of t in the domain of f. The smallest such number c is called the period of the function. 4-3

Your Turn Work Sheet 4.3 4-3

Home Work P. 381, #2, 6, 8, 18, 20, 22, 26, 30, 36, 40, 44, 48, 50, 52, 61-66, 68, 70 4-3