Transcript G. Measuring Angles

G. Measuring Angles
Math 30: Pre-Calculus
 PC30.1
 Extend understanding of angles to angles in standard
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position, expressed in degrees and radians.
PC30.2
Demonstrate understanding of the unit circle and its
relationship to the six trigonometric ratios for any angle in
standard position.
PC30.4
Demonstrate understanding of first- and second-degree
trigonometric equations.
Key Terms:
1. Angles and Measuring Angles
 PC30.1
 Extend understanding of angles to angles in standard
position, expressed in degrees and radians.
1. Angles and Measuring Angles
 So to convert between the two measurement systems we can
use the following to covertions.
 Degrees to radians
 Radians to degrees
 It is acceptable to omit the radians at the end. So instead of
2π/3 radians we can just write 2π/3 which is understood to
be radians.
Example 1
 When you sketch a 60º and 420º angle in standard position,
the terminal arms coincide. These are Coterminal Angles.
Example 2
 For example what are angles coterminal with:
 40º
 2π/3
Example 3
 All arcs subtending a right angle (π/2) have the same central
angle, but different arc lengths depending on the radius. Arc
length is proportional to the radius.
 This is true for any central circle
 Consider the 2 concentric circles.
 Radius of small circle is 1 and the larger circle is r.
 A central angle of θ rads in a subtended arc. AB on smaller
circle and CD on larger circle.
 We can write a proportion, when x represents the arc length
of the small circle and a arc length of the larger circle.
 Consider the circle with r=1 and central angle θ
 The ratio of the arc length to the circumference is equal to
the ratio of the central angle to one full rotation.
a=θr
 This formula, works for any circle, provided θ is in
rads and both a and r are in the same units.
 Radians are especially useful for describing circular motion.
 Arc length, a, means the distance travelled along the
circumference of a circle of radius r.
 For a central angle θ, in radians, a=θr
Example 4
Key Ideas
p.175
Practice
 Ex. 4.1 (p.175) #1-8 odds in each, 9,10, 11-13 odds in each,
14-22 evens
#3,5,6,7,9,11,12,13 odds in each, 15-27 odds
2. Unit Circle
 PC30.2
 Demonstrate understanding of the unit circle and its
relationship to the six trigonometric ratios for any angle in
standard position.
2. Unit Circle
 A unit circle is drawn on a Cartesian plane with the center at
the origin and has a radius of 1 unit (not necessarily a
Cartesian plane but that is how we will be using it)
 We can find the equation of the unit circle using teh
Pythagorean Theorem.
Example 1
Example 2
 The formula a=θr applies to any circle as long as a and r are
in the same units
 In the unit circle r=1 so the formula simplifies to a=θ(1)
or a=θ
 This means the central angle and its subtended arc on the
circle have the same value.
 You can use the function P(θ)=(x,y) to link arc length, θ, of
a central angle in a unit circle to the coordinates, (x,y) of the
point of intersection of the terminal arm and unit circle.
 If we join P(θ) to the origin we create a central angle θ in
standard position
 The central angle and arc length are both θ radians and θ
units respectively.
 Function P, takes real number values for central angles or arc
length on the unit circle and matches them with specific
points.
 For example, if θ=π, then point (-1,0). Thus, we write
P(π)=(-1,0)
Example 3
Key Ideas
p. 186
Practice
 Ex. 4.2 (p.186) #1-5 odds in each, 6, 7, 9-15 odds
#3-5 odds in each,6,7,9, 10-20 evens
3. Trig Ratios
 PC30.2
 Demonstrate understanding of the unit circle and its
relationship to the six trigonometric ratios for any angle in
standard position.
3. Trig Ratios
 For this section we will have to recall come prior knowledge.
 What are the three trig ratios that you know?
 Recall a unit circle has a radius of 1 unit.
 Also with a unit circle P(θ)=(x,y) where P is a point on the
circumference of the circle.
 There are 3 more Trig ratios that we have not looked at
before and they are called the Reciprocal Trig Ratios
 They are the reciprocals of sine, cosine and tangent.
 They are called cosecant, secant, and cotangent.
Example 1
 Exact values for the Trig ratios can be determined using the
special triangles and multiples of θ=0, π/6, π/4, π/3, π/2
or θ=0º, 30º, 45º, 60º, 90º for points P(θ) on the unit
circle.
Example 2
 We can also determine approximate values for primary trig
ratios using calculators. Most calculators can determine trig
values for angles in degrees or radians. You have to have your
calculator in the correct units.
 Most calculators can complete ratios for negative angles.
However, you can use your knowledge of the CAST rules to
determine if the ratio is positive or negative.
 You can find value of reciprocal trig ratios as well (csc, sec,
cot) using your calculators.
Example 3
 How do your find the measure of an angle when the value of
the trig ratio is given?
 Remember you calculator will only give you one answer
when there are probably 2. So it is best to use the Reference
angle and CAST Rule
Example 4
Example 5
Key Ideas
p.201
Practice
 Ex. 4.3 (p.201) #1-12 odds in each, 13-17 odds, 18
#1-12 odds in each, 13-23 odds
4. Intro to Trig Equations
 PC30.1
 Extend understanding of angles to angles in standard




position, expressed in degrees and radians.
PC30.2
Demonstrate understanding of the unit circle and its
relationship to the six trigonometric ratios for any angle in
standard position.
PC30.4
Demonstrate understanding of first- and second-degree
trigonometric equations.
4. Intro to Trig Equations
 Investigate – Trig Equations p.206
 In the investigation what you were doing was solving trig
equations. In the last section we were already solving very
simple trig equations.
 The same strategies will be used as when we used to solve
linear and quadratic equations
 The notation [0,π] means the same as 0≤θ≤π
 θε (0, π) means 0< θ< π
 θε [0, π) means 0 ≤ θ< π
 So the round brackets ( mean don’t include the end point
 The square [ mean include the end point
 Remember we can always check our answers because...
Example 1
Example 2
Example 3
Key Ideas
p. 211
Practice
 Ex. 4.4 (p.211) #1-3, 4-6 odds in each, 7-13, 16, 18
#3-7 odds in each, 8-22 evens