Transcript Chapter1:[2.1

Chapter 2
Trigonometric Functions
2.1 Degrees and Radians
2.2 Linear and Angular Velocity
2.3 Trigonometric Functions:
Unit Circle Approach
2.4 Additional Applications
2.5 Exact Values and Properties of
Trigonometric Functions
2.1 Degrees and Radians
Degree and radian measure of angles
Angles in standard position
Arc length and area of a sector of a circle
Radian Measure of Central Angles
 Example:
 Find the radian measure of the central angle
subtended by an arc of 32 cm in a circle of
radius 8 cm.
 Solution: q = 32cm/8cm = 4 rad
Radian-Degree Conversion Formulas
 Example:
 Find the radian measure of -1.5 rad in terms of p and in
decimal form to 4 decimal places.
 Solution: qd = (qr)(180º/p rad) =
 (-1.5)(180/p) = 270º/p = -85.9437º
Angles in Standard Position
Sketching angles in Standard Position
 Sketch these angles in standard position:
A. -60º
B. 3p/2 rad
C. -3p rad
D. 405º
Coterminal Angles
 Angles that differ by an integer multiple of 2 p or
360º are coterminal.
 Example:
Are the angles –p/3 rad and 2p/3 rad coterminal?
Solution: (-p/3) – (2p/3) = -3p/3 = -p No
 Example:
Are the angles -135º and 225º coterminal?
Solution: -135º - (-225º) = -1(360)º Yes
Area of a Sector of a Circle
A = ½ r2 q, r = radius and q = central angle
Example:
In a circle of radius 3 m find the area of the
sector with central angle 0.4732.
Solution: A = ½ 3m2(0.4732) ≈ 2.13 m2
2.2 Linear and Angular Velocity
Electrical Wind Generator
 This wind generator has propeller
blades 5 m long. If the blades are
rotating at 8 p rad/sec, what is the
angular velocity of a point on the
tip of one blade?
 Solution: V = 5 (8p) = 126 m/sec
2.3 Trigonometric Functions:
The Unit Circle Approach
Definition of Trigonometric Functions
Calculator Evaluation
Application
Summary of Sign Properties
Trigonometric Functions
The Unit Circle
 If a point (a,b) lies on the unit circle, then the following
are true for the angle x associated with that point:
 sin x = b
 csc x = 1/b (b ≠ 0)
 cos x = a
 sec x = 1/a (a ≠ 0)
 tan x = b/a (a ≠ 0)
 cot x = a/b (b ≠ 0)
Evaluating Trigonometric Functions
 Example:
Find the exact values of the 6 trigonometric functions for the
point (-4, -3)
The Pythagorean Theorem shows that the distance from the
point to the origin is 5.
 sin x = -3/5
 csc x = -5/3
 cos x = -4/5
 sec x = -5/4
 tan x = 3/4
 cot x = 4/3
Using Given Information to Evaluate
Trigonometric Functions
Example:
Given that the terminal side of an angle is
in Quadrant IV and cos x = 3/5 find the
remaining trigonometric functions.
b2 = 25 – 9 = 16, so b = 4
Sin x = 4/5, tan x = -4/3, csc x = -5/4,

sec x = 5/3 and cot x = -3/4
Reciprocal Relationships
Calculator Evaluation
 Set the calculator in the proper mode for each
method of evaluating trigonometric functions.
Use degree mode or radian mode.
 Example:
Find tan 3.472 rad
Solution: tan 3.472 rad ≈ .3430
 Example:
Find csc 192º 47’ 22”
Solution: csc 192º 47’ 22” ≈
1/ sin 192.7894… ≈ -4.517
Additional Applications
Modeling light waves and refraction
Modeling bow waves
Modeling sonic booms
High-energy physics: Modeling particle
energy
Psychology: Modeling perception
Light Rays
c1 sin 
=
c2 sin 
n=
speed of light in a vacuum
n
= 1
speed of light in a substance n2
c1 n1 sin 
=
=
c2 n2 sin 
Reflected Light
 Example:
 What is the angle of incidence  that will cause a light
beam to be totally reflected?
 Solution: sin  = (sin 90º)1/1.33

 = sin-1 (1/1.33)≈ 48.8º
2.5 Exact Values and Properties of
Trigonometric Functions
Exact values of trigonometric functions at
special angles
Reference triangles
Periodic functions
Fundamental identities
Special Angles
Using Special Angles for Points (a,b)
Example:
Find sec 5p/4
Solution: (a, b) = (-1/√2, -1/√2)
sec 5p/4 = 1/a = -√2
Example:
Find sin 135º
Solution: (a, b) = (-1/√2, 1/√2)
sin 135º = b = 1/√2
Angles on the Unit Circle
Using Special Angles for Points (a,b)
Example:
Find sin 7p/6
Solution: (a, b) = (-√3/2, -1/2)

sin 7p/6 = b = -1/2
Reference Triangle and Reference Angle
Reference Triangles and Angles
 Example:
Sketch the reference triangle and find the reference angle
 for q = -315º.
Solution:
Periodic Functions
 Adding any integer
multiple of 2p to x
returns the same
point on the circle.
 sin x = sin (x + 2p)
 cos x = cos (x + 2p)
 If sin x = 0.7714 then
sin(x + 2p) = 0.7714
Fundamental Identities
 csc x = 1/b = 1/sin x
 sec x = 1/a = 1/cos x
 cot x = a/b = 1/tan x
 tan x = b/a =
sin x / cos x
 cot x = a/b =
cos x / sin x
 sin2x + cos2x = 1
Use of Identities
Claim:
sin 2 x  cos 2 x
= cot x
tan x
Proof:
sin x  cos x
1
=
= cot x
tan x
tan x
2
2