Trigonometry & the Unit

Circle

Angles and Angle Measure Standard position: Positive angles Negative angles Up until this point, angles have been measured in __________________. One revolution is equal to ___________. Angles can also be measured in _______________. Radian Unit for radians

Degree Measure Radian (exact values) Radian (Approx. Values)

Converting Angle Measures A. DEGREES TO RADIANS Example: Convert each of the following to radians and sketch the angle:

a) 30°

b) 55°

c) 108− ° B. RADIANS TO DEGREES Example: Convert each of the following to degrees and sketch the angle:

a) 4π

b) 74π

−

c) 5.13

Coterminal Angles To find coterminal angles of a given angle, add or subtract ______ or ______. Example 1: Determine one positive and one negative angle measure that is coterminal with each angle. In which quadrant does the terminal arm lie?

a.) 50°

b.) 420− °

c.) 73π

d.) 54π−

Coterminal angles in General Form. Any angle given has an ____________________ number of coterminal angles. We can express this using the general form of coterminal angles: Radians Degrees: OR Integers (I or Z): Natural numbers (N): NOTE: Any variable may be used, as long as it is clearly identified. The variable, ______, is also commonly used. Example 2: Write an expression for all of the angles coterminal with each of the following angles where θ ∈ℜ .

a.) 130° b.) 34π

Example 3: For each angle, determine all angles that are coterminal in the given domain.

a.) 220° , 360 720θ− ° ≤ < °

b.) 8 , 4 43π π θ π− ≤ ≤

Arc Length of a Circle s = θ = r = Note: Remember, if there is no unit attached to the angle measure (ex. θ = 3.6), it is assumed to be in radians.

Example: Solve for θ. 15

Example Find the arc length of a bicycle wheel given a radius of 32 inches and a central angle of 50°

Example Albert the alien walks along the equator of his planet travelling 72.4 skips. If the diameter of the planet is 84.2 skips, through what central angle did Albert travel, to the nearest tenth of a degree?

θ

6

The Unit Circle Unit Circle Coordinates of a point, P, are (x, y). Coordinate of ( )P θ are (cos θ, sin θ) Why? Equation of the unit circle: or Determining Equations of Circles Any circle with centre at the origin at ( )0,0 is: Example 1: Determine the equation of the following circles with the origin as their centre:

1.) Radius of 5

2.) Radius of 9

This course will focus on the unit circle. Recall the unit circle has a radius of __.

Example 2: Determine whether the point 5 2 5,5 5

−

is on the unit circle.

Example 3: Determine the coordinates for all points on the unit circle that satisfy the conditions given and draw a diagram to support your answer.

a.) the x-coordinate is 23

.

b.) the y-coordinate is 22

− and the point is in quadrant III.

Recall quadrantal angles and their coordinates: Example 4: If ( )P θ is the point at the intersection of the terminal arm of angle θ and the unit circle, determine the exact coordinates of each of the following:

a.) 2

P π

b.) ( )2P π c.) ( )6P π d.) ( )35P π

Linking the Unit circle to the Special Angles

2π

4π

3π

6π

The Unit Circle

Recall the special triangles: Note: Example 5: If ( )P θ is the point at the intersection of the terminal arm of angle θ and the unit circle, determine the exact coordinates of each of the following:

a.) 6

P π

b.) 43

P π

c.) 34

P π −

Example 6: Identify a measure for the central angle θ in the interval 0 2θ π≤ ≤ such that ( )P θ is the given point for each of the following:

a.) ( )1,0− b.) 3 1,2 2

−

c.) 2 2,2 2

−

Example 7: Given that ( )P θ =

−

54,

53 ,

a.) Find ( )πθ +P b.) Find ( )πθ −P

c.) Find

+

2πθP d.) Find

−

2πθP

Summary:

Trigonometric Ratios We are already familiar with the 3 primary trig ratios: Reciprocal Trigonometric Ratios: Calculating Trigonometric Values for Points Not on the Unit Circle: (this is a review from grade 11, except we now know the reciprocal trig ratios!) Example 1: A.) The point A ( )5, 12− − lies on the terminal arm of an angle in standard position. What is the exact value of each trigonometric ratio for θ ? B.) Determine the corresponding point ( )P θ on the unit circle.

Determining the Trigonometric Ratios for Angles on the Unit Circle

Example 2: The point A 1 2 2,3 3

−

lies at the intersection of the unit circle and

the terminal arm of an angle θ in standard position. Draw a diagram to model this situation and determine the values of the six trigonometric ratios. Example 3: Determine the exact values of the other five trigonometric ratios under the given conditions.

a.) 1tan , 23

θ π θ π= − ≤ ≤

Exact Values of Trigonometric Ratios To determine exact values of trig ratios, we use the ____________________ and

the _____________ rule. The exact values are also found on the unit circle as x-

and y-coordinates.

Example 4: Determine the exact value for each of the following:

a.) 11sin6π

b.) 2cos3π −

c.) tan2π

d.) cot2π

e.) csc 225° f.) ( )sec 300− °

g.) 4 5cos csc3 6π π •

h.) 2

2 5 2sin cos4 3π π +

Approximate Values for Trigonometric Ratios Example 5: Determine whether each trigonometric ratio is positive or negative without using a calculator, then, using a calculator, determine the approximate value for each correct to four decimal places.

a.) 7tan5π

b.) sec6.5 c.) sin 254° d.) ( )csc 70− °

e.) 4sec9π

Solving Trigonometric Ratios & Equations Set notation and Interval notation. So far we have been expressing the domains (intervals) in set notation. We may also use interval notation. For and≤ ≥ we use [ ] brackets For < and > we use ( ) brackets Set notation Interval notation 0 2θ π≤ ≤ π θ π− ≤ <

0 180θ° < ≤ ° [ ]0,π ( 180 ,360 ]− ° ° Finding Angles using Trig Ratios (Solving for θ) Steps: 1.) Find the reference angle, rθ .

2.) Determine the quadrants using the CAST rule. 3.) Calculate θ in each applicable quadrant. 4.) Pay close attention to the domain (interval) given for θ .

Recall how to use rθ for each quadrant: Degrees Radians

Example 1: Determine the measures of all angles that satisfy the following. State exact answers when possible, otherwise state answers correct to 3 decimal places. (This means _______________________.)

a.) 3sin , 0 22

θ θ π= ≤ ≤

b.) [ ]cot 1, 0,2θ π= −

c.) [ ]sec 2.4, 0,θ π=

d.) ( )csc 1.75, 0,2θ π= −

e.) [ ]3sec , 0,24

θ π=

Verifications You can ___________ your solutions by ____________________ the angles back in to the equation and evaluating both sides of the equation.

Example 2: Verify to determine if 53π is a solution to 2sec 2sec 0x x− =

Trigonometric Equations Trigonometric equations are equations involving trigonometric ratios. We use the same algebraic techniques used to solve linear and quadratic equations to solve trig equations. Let’s look at isolating the variables and trig functions. Algebraic equation Trigonometric equation. 1.) 2 1 0x + = 2cos 1 0θ + = 2.) 2 0x x− = 2sin sin 0θ θ− = 3.) 2 6 0x x+ − = 2tan tan 6 0θ θ+ − = 4.) 210 3 4 0x x+ − = 210cos 3cos 4 0x x+ − = Examples: Solve each trigonometric equation in the specified domain. State exact answers where possible, otherwise give approximate answers to the nearest thousandth.

a.) 5sin 2 1 3sin , 0 2θ θ θ π+ = + ≤ ≤ (Verify this one.)

b.) 2sec 2sec 0, 0 2x x x π− = ≤ ≤

c.) [ ]2tan 5 tan 4 0, 0,2θ θ π− + =

d.) [ ]2 1sin , 0,24

θ π=

e.) [ ]23sin 6sin 2 0, 0,2θ θ π+ + =