Grade 12 Pre-Calculus Mathematics Notebook …...Grade 12 Pre-Calculus Mathematics Notebook Chapter 4 Trigonometry and The Unit Circle Outcomes: Ti, T2, T3, T5 12P.T.1. Demonstrate

Grade 12 Pre-Calculus Mathematics

Notebook

Chapter 4

Trigonometry and The Unit Circle

Outcomes: Ti, T2, T3, T5

12P.T.1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

12P.12. Develop and apply the equation of the unit circle.

12P.T.3. Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees.

12P.T5. Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians.

Page 1

4.1 Angles and Angle Measure

Ti (p.166-174)

An angle in standard position has its centre at the origin and its initial arm along the positive x-axis.

There are positive and negative angles.

h

Positive Angles Counter-clockwise

Negative Angles Clockwise

Page 2

Radian Measure of an Angle

• The formula for the circumference of a circle is C = 27tr • The unit circle has a radius = 1 • Therefore, the circumference of the unit circle is 2n

2n = 6.283185...

This means that the distance traveled from the initial arm all around the circle and back again is 6.283185...

Revolutions Degrees Radian Measure

1 revolution 3 (0 o° a- 7 radians 6.283185... radians

i — revolution 2

kW' Ti radians 3.141592... radians

1 — revolution 4

■WR, q 0`'

tlopel .-- radians 1.570796... radians

1 revolution 1 c 71- 0.017453... radians 11,--0 radians 360

Note that 1 radian = (Boj o „...„ 57.30

Page 3

CONVERTING DEGREES TO RADIANS I k

-Tr (t.---o

CONVERTING RADIANS TO DEGREES

Nk. F-up 6") •

'TC

Ex.1: Express the following angle measures in radians.

a) 30° b) 225°

a as- / ■cisv

5- a

c) 720°

-2_ 05-1)

atm LtTr

Ex.2: Express the following angle measures in degrees.

a)

_I(

b) 1.6 c) 57r 6

1 . cro Ssl1 (290 11--)

o3

koq 64 - s-o

Page 4

The central angle is the relationship between the length of the arc and the radius of the circle.

0 = central angle (*must be in RADIANS!)

S = arc length (also referred to as a)

r = radius

The equation that represents this relationship is:

S=Or

Note: If there is no unit attached to the angle measure (ex. 0 = 2.5) it is assumed to be in radians.

Ex3: A bicycle tire has a radius of 0.5m and travels a distance of 1.5m. Determine the rotated angle, in degrees.

e

-e ,s--

3 0

e- • SL-10

Homework: Page 175 #1-4, 6, 12, 13, Cl, C2

Page 5

4.2 The Unit Circle — Part 1

The unit circle with radius 1 is centered at the orgin.

T2 (p.180-186)

We use the notation P(o) to indicate a point on the circle.

Vs--4-v-c-stesigtb. P(8) = defined by a point (x,y)

Since the radius is 1, then the equation of the unit circle is x2 + y2 =1.

Page 6

Exl: A point (-2 ,y) is on the unit circle. Find the value of y. 3

-7_ X A- L-1\

`‘)

Note: There are 2 possible values for y since there are 2 points on the unit 2

circle that share the coordinate x = —3 .

Ex2: Determine whether or not the point (-25 ,-35) is on the unit circle. Justify

your reasoning.

as

"3

S-

Page 7

PD) CO SG

7 La)

C S

7_ (c) —

--- 3(0 -2_

Similar Triangles A Ex3: The point P(0) lies on the intersection of the unit circle and a line

joining the origin to the point (4, 3). Find the coordinates of P (0) .

6)

Ex4: The point P(0) lies on the intersection of the unit circle and a line joining the origin to the point (-3, 6). Find the coordinates of P (0) .

Page 8

Finding missing circular functions for 0 when one function is given.

Ex5: Find cos 9 and tan 9 over the interval —37r 27r when sin 9 = --3

2 5

4;--)r■ e- --=----

a -4-- -v-\o-z-

Coterminal Angles

Coterminal angles share the same terminal arm, such as 70°, 70° + 360° = 430° and 70° – 360° = –290°.

700

The coterminal angle can be found by adding or subtracting revolutions; either ±360° when given degree measure or ±27z- when given radian measure. There are an infinite number of coterminal angles.

Page 9

(N,

Ex6: a) Determine 3 coterminal angles for 40°.

(AC -k- to ck.0 0

o

- -s ao

b) Determine 3 coterminal angles for 6 -Tr a cc

c) Find a coterminal angle to 740° over the, interval -360° < 0 < 0°.

-Trey) 1(.0 0 --z_ cad 360.

-

d) Express a general form for all coterminal angles of jr-5 3

e)Determine all coterminal angles of over the interval 3

[-43r, 270.

3

n

-k--

LAW, STc- VB.) fit

5 3

Homework: Page 175 #7-11, C3 Page 186 #2 & 3 Page 202 #8 & 13

Page 10

4.2 The Unit Circle — T2, T3

(p.180-186)

I.._ .._ .._ .._ .._.i

' The 45° Triangle : •1_ .. _ .. _ .. _ .. _ .. J

i -■•••*"..----

\r-2-

: The 30°-60°-90° Triangle ! L .. _ .. _ .. _ .. _ .. _ .. _ .. — .. -.

-)

Page 11

The Unit Circle

Homework: Page 187 #4-6, 12, 16, Cl , Page 12

The Unit Circle

P = (x, y) P(9) = (co se , sine)

X.2 + y2 = 1 cos 20 + sin2 0 = 1

Page 13

4.3 Trigonometric Ratios

Recall: e = arc length P(e) = defined by a point (x, y)

T3 (p.192-201)

If we use the trigonometric ratios SOH CAH TOA, then

sin 0 = 1 —> sin 0 = y 1

cos 9 = —x

—> cos 0 = x 1

,-) Y sin 9

tan u = — —> tan 9 =

x cos 0

Thus, any point on the unit circle can be described as

P(0) = (cos0, sin0)

Primary Functions Reciprocal Functions

sin 0 = y 1 1 cosecant esce = = sin 0 y

secant sec 0 = 1 1=

cos 0 x

tan 0 = —y

x , 1 x cos° cotangent cot u= = =

tan 0 y sin 0

Page 14

— r'S "

6

4S-1 G — '3

12& -C2.7aCSC er — 1-3 \ -2-

-1 b) Find all 6 trigonometric ratios for 0.

13

C %

ete\-c.

Exl: The point (-5 --12) lies on the erminal an of an angle 0 in standard 13' 13

position.

a) Draw a diagram to represent this situation.

1.,C45t.

k‘ A

Ex2: The point (-3 4) lies on the4erminal arm of an angle 0 in standard 5 5

position.

a) Draw a - 3

to represent this situation.

b) Find all 6 trigonometric ratios for 0.

3

Page 15

4.3 Determining Exact Values T3 (p.192-201)

Ex 1: Determine the exact value of the following trigonometric ratios:

n- a) cos —

3 b) sec — =

3

c) sin (

6)

0 d) cos-7g . 4

27z- e) cot(270°)

0 CSC

3 \FS

g) tan-112z•

= 4

h) sec(57r) =

Page 16

Ex2: Determine the exact value of the following expressions:

a) cos(120°)—tan(-135°)

( 77z- b) sin 2(-77-t-

j+cos 2 6 6

-t-

L

( 3)-c 7-C c) Cot — — + CSC —

\ 4 ) 2

tan2( — TC\

sec(

471-\ \ 3)\3)

Homework : Page 201 #1, 2 (choose 6), 3, 6, 9 Page 17

4.4 Introduction to Trigonometric Equations T5 (p.206-211)

Exl: Solve the following equations over the indicated domain.

a) 2 cos 0-3 = —1, 0° < 540°

co(3 ------ 1

0

b) 4secx+8=0, 0_x<27z-

' Li

Co -oat). — 2_

aTc- it

-23

c) sin2 x — sin x = 0, 0° x 180°

d) tan28 — 1 = 0, —7C < 9 < it

--1) ----

S Tv Stc‘

—3 rr- –11 _- Tr 3ir t gl ---̀1 —71

e) (2sinx + 1)(2sinx — = 0, 0 x

-- S'11(\_ •-)

7i[ 1 cr ) 211-

3

-5

Page 18

0 o

01,U_ ,>+

LC \

f) tan2 x – tan x + 6=0, 1800 x360°

CA \IL 11' (_(0

— – S _

g) sin2 x + sin x – 12 =0, 0_x__271-

C ,̀5 )(±t t )

5-1‘ N S 0 (

h) csc2 x + csc x –12 = 0, 0 x 27z-

C >c —

C-3 c_ = -s

-

x=- ")) (-)'

= (0

)

Page 19

General Solution If the domain is real numbers, there are an infinite number of rotations on the unit circle in both a positive and negative direction.

To determine a general solution, find the solutions in one positive rotation. Then use the concept of coterminal angels to write an expression that identifies all possible measures.

There are different ways to request general solution answers. They are: • domain is all real numbers • x E R or 9 E R

• general solution

1 Ex2: a) Solve cot = v over the interval — 27c 9 27t

-3

ilkAn. €3

t — — S L

3

b)Ift9ER.

-NV- 4- aTrv 5

--t an- e-\

Page 20

Ex3: a) Solve tan 0 = -4 over the interval - 1f s 0 s 1f .

e ( � -t �---l

l L{ )

er =

b) If the domain is all real numbers.

Ex4: a) Solve cos P = 0 over the interval -41f s P s 21f.

v ·-

l\ 2> -rt --n- - �\\.-

?..-- "'2---z:._

b) Find the general solution

- �

-z.__

Homework: Page 211 #1-3, 4 & 5 (choose 3), 7-11, 13, 18, C4, C3

Page 21

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c_

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