The document discusses the unit circle and trigonometric functions. It provides information on: - The initial and terminal sides of angles on the unit circle - The values of sine, cosine, and tangent in each quadrant of the unit circle - Using Pythagorean theorem to determine values on the unit circle - Definitions of sine, cosine, and tangent in terms of the x- and y-coordinates on the unit circle - Rules for transforming basic sine and cosine functions - Steps for solving trigonometric equations by taking the inverse sine or cosine of both sides.

1. Unit Circle
Getting circles, trig, triangles
and radians all together for
over 100 years

2. Solving Equations
 To find zeroes:
 y = sinx = 0
 Solve sin x = 0
 Take inverse sine of both sides to
solve for x
 So x = 0 and π
 T

3. The Unit Circle

4. Unit Circle
 Initial side: the right half of the x
axis, equal to the unit circle radius, is
considered the starting point for any
angles in the unit circle.
 Terminal side: the side, equal in
length to the initial side, after a
rotation of angle Θ.

5. Unit Circle Quadrant Table
Arc
Measure
Quad Sign
Sin Θ
Sign
Cos Θ
Sign
Tan Θ
0< Θ < π/2
π/2 < Θ < π
π < Θ < 3π/2
3π/2 < Θ < 2π

6. Pythagoras
 Pythagoras can help determine some
of the values on a unit circle.
 E.g. At P(Θ) = P(30˚) or P(π/6)
 The hypotenuse is 1, the opposite is 0.5
by the 30˚ rule, so what is the adjacent
side’s length?
 Use Pythagoras’ Theorem to solve.

7. Unit Circle Trig
 P(Θ) is the trigonometric point on the unit
circle after rotation of angle Θ
 sin Θ = the y coordinate of P(Θ)
 cos Θ = the x coordinate of P(Θ)
 tan Θ = sinΘ = y coordinate of P(Θ)
cosΘ x coordinate of P(Θ)

9. Activities
 P. 238, Q. 1,2,5, 6, 7, 8

10. Sine Function: y = sin x

11. Terms
 Amplitude = how much the y
increases and decreases from the
midpoint.
 Amplitude = (max-min)/2
 Amplitude is always positive = A
 In the equation amplitude is “a”
which can be positive or negative
 So A = |a|

12. Terms
 Period, p, describes the length of one
full cycle or the wavelength of the
function.
 Period is determined from the term
“b”.
 p = 2π/|b|, p is positive
 The longer the period, the lower the
frequency.
 Period = 1/frequency = 1/2π for sine fn

13. Rule for Basic Sine Function
Property Basic Function Transformed Function
Rule y = sin x y = a sin b(x-h) + k
Graph Sinusoidal
a = 1
Period =2π
Sinusoidal
a = amplitude
p = 2π/p p = period
h = horizontal translation
k = vertical translation
Domain ¶R ¶R
Range [-1,1] [k + A, k – A]

14. Rule for Basic Sine Function
Property Basic
Function
Transformed Function
Extremes Max 1
Min -1
Max k+ A
Min k - A
Variation Periodically
Inc or dec
Periodically increase or
decrease
Sign Depends Depends
Inverse Not a fn Not a fn

15. Exam QuestionThe population growth P of the native people of Northern Quebec is represented by the graph of
the sinusoidal function with equation
    6004
4
sin400 

 ttP
where t is the number of years elapsed since 1984.
Which one of the graphs below corresponds to this situation?
A)
1 000
600
200
2 4 6 8 t
E(t) C)
1 000
600
200
2 4 6 8 t
E(t)
B)
1 000
600
200
2 4 6 8 t
E(t) D)
1 000
600
200
2 4 6 8 t
E(t)
BIM : Information
Origin : BIM National Committee - Montérégie Region

16. Activities
 Page 253, Q. 6,9,13
 N.B. Many answers are periodic, i.e.
repetitive , (like me). So an answer
of 1 with a period of π, would give a
set {1, 1+ π, 1+2 π, 1+3 π….}
1 + n π, n Є Z, where n is an integer

17. Cosine Function: y = cos x

18. Rule for Basic Cosine Function
Property Basic Function Transformed Function
Rule y = cos x y = a cos b(x-h) + k
Graph Sinusoidal
a = 1
Period = 2π
Sinusoidal
a = amplitude
b = 2π/p p = period
h = horizontal translation
k = vertical translation
π/2 out of phase of sine
Domain ¶R ¶R
Range [-1,1] [k + A, k – A]

19. Rule for Basic Cosine Function
Property Basic
Function
Transformed Function
Extremes Max -1
Min 1
Max k + A
Min k - A
Variation Periodically
inc or dec
Periodically inc or dec
Sign Depends Depends
Inverse Not a fn Not a fn

20. Exam Question
If the value of x varies from  to 2, the function f(x) = cos x
A) increases in [, 2].
B) decreases in [, 2].
C) increases in 


 

2
3
, and decreases in .2,
2
3






D) decreases in 


 

2
3
, and increases in .2,
2
3







21. Activities
 Page 263, Q. 4,6,9, 14

22. Tangent Function y = tan x

23. Rule for Basic Tangent Function
Property Basic Function Transformed Function
Rule y = tan x y = a tan b(x-h) + k
Graph Sinusoidal
a = 1
asymptote
Sinusoidal
a = amplitude
b = π/p p = period
h = horizontal translation
k = vertical translation
Domain ¶R ¶R
Range [-1,1] [k + a, k – a]

24. Rule for Basic Tangent Function
Property Basic
Function
Transformed Function
Extremes None None
Variation Depends Depends
Sign Either inc or
dec per fn
Either inc or dec per fn
Inverse Not a fn Not a fn

25. Exam Question
Which one of the following graphs represents the tangent function for the interval [0, ]?
A) f(x)
x

2
C) f(x)
x

B) f(x)
x

2
D) f(x)
x


26. Activities
 Page 270, Q. 4,9,10

27. Solving Trig Equations
 To solve sine equations for zero or
any other number, take the following
steps:
 Y = sin x = 0
 Sin x = 0
 Take inverse sine on both sides to
solve for x
 Based on the unit circle that gives
two possible zeroes.

28. Unit Circle & Equations
 So, sin x = 0
 The inverse sine of (0) gives 2
results
 sin -1 (sin x) = sin -1 (0)
 So x = 0 and x = π (180˚)
 Why?
 Because sin 0˚ = 0 and sin 180˚ = 0.
Remember sin Θ = sin (180˚ – Θ)

29. Solving Sine Equations
 E.g sin (x-2) = √2/2
 sin -1 (sin (x-2) = sin -1 (√2/2)
 So (x-2) = 0.785 or (x-2) = 2.356
 x = 0.785 +2 cuz sin(x)=sin(π-x)
 x = 2.785 so x = 4.356
 So the answers are 2.785 and 4.356
 To allow for a periodic/repeating answer
 2.785 + 2πn, where n is an integer
 4.356 + 2πn, n Є Z


30. Cosine Equations
 For cosine, remember that
 cos (x) = cos (-x) = cos (2π-x)
 E.g. cos (x)= 0
 cos -1 (cos (x)) = cos -1 (0)
 x = π/2 or x = 3π/2

31. Solving Cosine Equations
 E.g cos (2x+3) = √3/2
 cos -1 (cos (2x+3) = cos -1 (0.866)
 So (2x+3) = 0.523 or (2x+3) = 5.757
 2x = 0.523 -3 cuz cos(x)=cos(2π-x)
 x =-2.477 so x = 1.3785
 So the answers are -2.477 and 1.3785
 To allow for a periodic/repeating answer
 -2.477 + 2πn, where n is an integer
 & 1.3785 + 2πn, n Є Z