4.1: Angles & Angle Measuredobsonmath.weebly.com/uploads/1/1/8/0/11809374/pc_12_ch._4_-_not… · 4.1: Angles & Angle Measure ... (in degree) or 2 S (in radian). ... We saw above

Math 12 Pre-Calculus Chapter 4: Trigonometry and the Unit Circle

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4.1: Angles & Angle Measure In Trigonometry, we use degrees to measure angles in triangles. However, degree is not “user friendly” in many situations (just as % is not user friendly unless we change it into decimals.)

In this unit, there is a better way to measure angles: radian .

Consider a unit circle. With each circle having 360o and circumference being

C = , there is a relationship between the angle and its circumference:

angle circumference ------------------------------------------ 360o

That means, 2 (radian) can now be used to describe the angle 360o (degree). Similarly, with a half-circle, we have:

angle circumference ------------------------------------------ 180o

When working with radians we don’t need to include the units. If an angle measure has no units, we can assume it is in radians.

To convert between units, we will use unit analysis, as in Science with the conversion 180 180o . Ex.1: Convert the following angles.

a) 40o in radians b) 1 radian in degrees 180o 40o x ---------- 1 x Then, 90o x ------------ = ---- in radian 180o 2 We use arc length to describe a portion of the circumference. To calculate the arc length of a circle, we need to convert angles between degrees and radians using the conversion factor:


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OR (arc length) = (angle in radian) x (radius) Ex.3: Find the unknown.

a) Given a = 28 and r = 5, find . b) Given r = and = 3

, find a .

.9o = 120o x ------- = 2.0945 180o

Recall from last year that a standard position angle is one measured from the positive x-axis to a terminal arm. Angles measured counter-clockwise are considered positive, and angles measured clockwise are considered negative. The circle below is divided by lines with different angles. Label the angles in both degrees and radians. (Hint: The dotted lines cut the 90o in halves, and the shadow lines divide the 90o into 3 equal parts.)

Ex.1: Draw the angles.

a) 30o and 390o b) 4

3,

4

5


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Angles can go around the circle more than once before stopping on the same terminal. Co-terminal angles are angles in standard position that share the same terminal (stopping point). They can be found by adding or subtracting multiples of 360o (in degree) or 2 (in radian). Ex.2: Determine one positive and one negative angle measure that is co-terminal with each angle. In

which quadrant does the terminal arm lie?

a) b) c)

Co-terminal Angles in General Form: We saw above that we can find any co-terminal angle by adding or subtracting any number of complete revolutions ( or ) of a circle. In general, we can write any co-terminal angle as: In degrees: In radians: Ex.3: Graph each angle below. Then write a general equation to express its co-terminal angle.

a) = 210o b) = 6

40

430

8

3

360

2


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4.2: The Unit Circle & The Special Triangles The unit circle is a circle with radius 1 centered at the origin (0,0). So the equation for a circle with centre (0,0) and radius r is: Ex. 1: Determine the equation of a the circle with centre at the origin and radius 2. Ex. 2: Determine the coordinates for all points on the unit circle that satisfy the conditions given. Draw a diagram in each case.

a) the x-coordinate is b) the y-coordinate is and the point is

in quadrant III

2

3

1

2

y

x

1

P(x, y)

o

What is the equation of the unit circle?


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We will use the notation to show the relationship between an angle in standard

position and its corresponding point P on the unit circle. For example, if , then we could write .

Special Triangles: Recall the 2 special triangles from last year (below). We will need to use these to discuss coordinate points on the unit circle – for this we will need the special triangle angles in radians.

Ex. 3: a) On a diagram of the unit circle, show the multiplies of in the interval .

b) What are the coordinates for each point in part a)?

P() (x, y)

P( ) (1,0)

3

0 2

P()


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4.3 (part 1): Reference Angles & The Primary Trig. Ratios Recall: For any angle in standard position, its reference angle is the acute angle between its

terminal arm and the x-axis.

To reduce the complexity of angles, reference angles are used for trig calculation. ----------------------------------------------------------------------------------------------------------------------------------- Before we spend more time on the different trigonometric ratios (e.g. sine, cosine, tangent, etc…), it is important to know (and be familiar with) which of the four quadrants a given standard angle is located – especially when the angle is in radian. Ex.1: Draw each angle. Then, state the quadrant in which it is located.

a) 7

2 b) 3.5 c)

3

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Ex.2: Draw each angle. State its reference angle AND the formula for its coterminal angles.

a) 120o b) 6

c) 3.5

R

R R

C

C C


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The following are called the primary trigonometric ratios: we use SOH CAH TOA. In the diagram below, let’s apply SOH CAH TOA

sin opposite side

hypotenuse

cos adjacent side

hypotenuse

tan opposite side

adjacent side

Ex.3: Draw each angle below. Then, evaluate using the special triangles (no calculator!) a) sin 120o b) tan ( /3)

c) sin (-5 /4) d) cos 480o

Axial Angles (angles on the x-axis or y-axis):

y

x

r

P(x,y) y)

I

IV III

II

o


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4.3 (part 2): Secondary Trig. Ratios The following are called the secondary trigonometric ratios. They are the reciprocals of the primary trigonometric ratios. Cosecant Secant Cotangent

csc θ = sec θ = cot θ =

Ex.1: The point (4, -3) is on the terminal arm of an angle θ. a) Draw θ in its standard position.

b) Calculate θ.

c) Find all six trig ratios in exact values.

sin csc θ =

cos sec θ =

tan cot θ =

Recall the method of remembering which quadrant is positive/negative for each trig. ratio. A All S A

S Students Sine All T Take C Calculus T C

Tangent Cosine

Ex.2: Given the following conditions, where would be?

a) cos > 0 & tan < 0 b) sin < 0 & cot > 0


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Ex.3: Find sec θ in exact value if θ is in

quadrant II and cot θ =5

12 .

Ex.4: If x is on a terminal arm, find the

measure(s) of angle x if sin x =1

5.

Ex.5: Determine each of the trigonometric ratios. Draw a diagram to support your answer. a)

cos260 (nearest hundredth) b)

csc(70) (nearest hundredth)

c) sin135 (exact value) d) 6

5cot

(exact value)

Ex.6: Determine the measures of all angles that satisfy the following. Use diagrams to support your

explanations.

a) cos 0.366 where 0 360 b) 2

sec3

where 2 2


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4.4 (part 1): Trigonometric Equations (first and second degree)

** When solving Trigonometric Equations, DRAW a picture first **

Ex.1: If 0 2 and sec 2 ,

solve for in exact value.

Ex.2: Solve for if 20

and2

2sin .

Solving for :

1) Identify the correct quadrant(s) – use ASTC Draw a Picture!

2) Find the reference angle (R ) - use SOH CAH TOA and the Special Triangles.

3) Use R to find

1 and 2 - use addition/subtraction.

** When solving for an angle, ignore the negative sign – it has already done its job by finding the right

quadrants! ** Answer in exact values whenever possible. (use special triangles)

Ex.3: Solve 05tan3 for .

0 3600 0


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Second Degree Trig. Equations Solve as you would a Quadratic Equation (set = 0, factor). Then solve the factors individually.

Ex.4: Solve cos3cos2 2 for

20 .

Ex.5: Solve 2sin3sin 2 xx for 0 180x

4.4 (part 2): Trigonometric Equations (general solutions) Because trigonometric functions are periodic, they repeat themselves.

Sine & cosine repeat themselves every 2 (360o), and tangent repeats for every (180o).

When solving a Trigonometric Equation for all real solutions ( x ), we need to include all

co-terminal angle solutions: i.e. 2x n , n Sometimes we can reduce the general solution if an angle is repeated within the circle (eg.

every half circle). Ex.1: Solve sinx = -1 for all real solutions. Ex.2: Solve tan x = 0 for all real solutions.


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Ex.3: Solve over the set of all real numbers.

a) sec2 2 (in radians) b) tan3 1 (in radians)

c) 2tan 5tan 4 0 (in radians)

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4.1: Angles & Angle Measuredobsonmath.weebly.com/uploads/1/1/8/0/11809374/pc_12_ch._4_-_not… · 4.1: Angles & Angle Measure ... (in degree) or 2 S (in radian). ... We saw above