G. Measuring Angles Math 30: Pre-Calculus
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.
1. Angles and Measuring Angles • PC30.1 • Extend understanding of angles to angles in standard position, expressed in degrees and radians.
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.
When you sketch a 60º and 420º angle in standard position, the terminal arms coincide. These are Coterminal Angles.
For example what are angles coterminal with: • 40º • 2π/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 θrads 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
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.
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.
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)
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.
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/2or θ=0º, 30º, 45º, 60º, 90ºfor points P(θ) on the unit circle.