445 102 Mathematics 2 Module 4 Cyclic Functions

445. 102 Mathematics 2 Module 4 Cyclic Functions Lecture 2 Reciprocal Relationships

445. 102 Lecture 4/2 Administration Last Lecture Looking Again at the Unit Circle Some Other Functions Equations with Many Solutions Summary

Administration Chinese Tutorials Text Handouts Modules 0, 1, 2 —> p 52 Module 3 —> pp 87 - 109 Module 4 —> pp 77 - 88 This Week’s Tutorial Assignment 4 & Working Together

445. 102 Lecture 4/2 Administration Last Lecture Looking Again at the Unit Circle Some Other Functions Equations with Many Solutions Summary

Radians A mathematical measure of angle is defined using the radius of a circle. 1 radian

sin(ø) 1 ø sin(ø)

Post-Lecture Exercise 1 45° = π/4 radians 60° = π/3 radians 80° = 4π/9 radians 2 full turns = 4π radians 270° = 3π/2 radians 2 3 π radians = 180° 3 radians = 171. 9° 6π radians = 3 turns f(x) = sin x is an ODD function. 4 f(2. 5) = 0. 598 5 6 f(20) = 0. 913 f(– 4) = 0. 757 f– 1(0. 5) = 0. 524 f– 1(0. 3) = 0. 305 f– 1(– 0. 6) = – 0. 644 The domain of f(x) = sin x is the Real Numbers The domain of the inverse function is – 1 ≤ x ≤ 1 f(π/4) = 0. 707

Lecture 4/1 – Summary There are many functions where the variable can be regarded as an ANGLE. One way of measuring an angle is that derived from the radius of the circle. This is called RADIAN measure. From the UNIT CIRCLE, we can see that the SINE of an angle is the height of a triangle drawn inside the circle. Sine(ø) then becomes a function depending on the size of the angle ø.

The Sine Function (Many Rotations)

Preliminary Exercise

445. 102 Lecture 4/2 Administration Last Lecture Looking Some Again at the Unit Circle Other Functions Equations with Many Solutions Summary

cos(ø) 1 ø cos(ø)

Constructions on the Unit Circle 1 tan(ø) sin(ø) ø cos(ø)

The Cosine Function (Many Rotations)

The Tangent Function (Many Rotations)

445. 102 Lecture 4/2 Administration Last Lecture Looking Again at the Unit Circle Some Other Functions Equations Summary with Many Solutions

The Secant Function

sec ø/ = sec ø = 1/ 1 cos ø sec ø 1 cos(ø) 1

Inverse Functions The sine function maps an angle to a number. e. g. sin π/4 =0. 707 The inverse sine function maps a number to an angle. e. g. sin-10. 707 = π/4 Note the difference between: The inverse sine: sin-10. 707 = π/4 The reciprocal of sine: (sin π/4)-1 = 1/(sin π/4) = 1/0. 707 = 1. 414

Inverse Functions Here is a quick exercise. . (remember to give your answers in radians): 1. 2. 3. 4. 5. 6. What angle has a sine of 0. 25 ? What angle has a tangent of 3. 5 ? What angle has a cosine of – 0. 4 ? What is sec π/2 ? What is cot 5π/3 ? What is arctan 10 ?

445. 102 Lecture 4/2 Administration Last Lecture Looking Again at the Unit Circle Some Other Functions Equations Summary with Many Solutions

An Equation 2 cos ø – 0. 6 = 0 2 cos ø = 0. 6 cos ø = 0. 3

A Special Triangle 1 unit

A Special Triangle 1 1

A Special Triangle √ 2 1 π/ 4 1

A Special Triangle sin π/4 = 1/√ 2 cos π/4 = 1/√ 2 1 tan π/4 = 1/1 = 1 π/ 4 1

Another Special Triangle 2 units

Another Special Triangle 2 √ 3 1

Another Special Triangle π/ 6 2 √ 3 π/ 3 1

Another Special Triangle sin π/6 = 1/2 π/ cos π/6 = √ 3/2 sin π/3 = √ 3/2 6 cos π/3 = 1/2 2 √ 3 tan π/6 = 1/√ 3 π/ 3 1 tan π/3 = √ 3/1 =√ 3

445. 102 Lecture 4/2 Administration Last Lecture Looking Again at the Unit Circle Some Other Functions Equations with Many Solutions Summary

Lecture 4/2 – Summary Sine, cosine and tangent can be seen as lengths on the Unit Circle that depend on the angle under consideration. So sine, cosine and tangent are functions where the angle is the variable. For each of these there is a reciprocal function. The graphs of these functions can be used to “see” the solutions of trigonometric equations

445. 102 Before Lecture 4/2 the next lecture. . . . Go over Lecture 4/2 in your notes Do the Post-Lecture exercise p 84 Do the Preliminary Exercise p 85 See you tomorrow. . . .