Name:
Date:
Tuesday 7th May 2019
Teacher:
Mr Innes
Unit 1 Maths Methods
Chapter 14: Circular Functions
Question List
14A
14B
14C
14D
14E
14F
14G
14H
1 b f, 2 g h, 3 d h, 4 d h, 5 a c g, 6 e f, 7 iii iv
for a b c
1 a c e g, 2 d h, 3 c f h
1 d e f, 2 f g, 3 a c e
1aceg
1 b d f h, 2 b c, 3, 4, 5, 6 RHS, 7 c f
1 a f h j, 2 g h l, 3 b d f h
1 c f l, 2 b d, 3 RHS, 4a, 5 c d, 6, 7 a e
1 a c, 2 c f, 3 c f, 4 a, 5 a b, 7 e f, 8 a d
Required for Outcome 2 and
Mid-Year Exam
14I
14J
14L
14K
14L
14M
14N
14O
1 a c f i, 2, 4
1cd 2acd
1 b c, 2 b c, 3 a c, 4ac
1 b d f h j, 2 c d, 5
1 b c, 2 b c, 3 a c, 4ac
1 c d, 2 c
1, 2, 4
3, 5, 7, 8
Required after the Mid-Year
Exam
1
Exercise 14A: Measuring Angles in Degrees and Radians
Unit circle
The diagram shows a unit circle, i. e. a circle of radius 1 unit.
The circumference of the unit circle = 2𝜋 × 1 = 2 𝜋 units
Radians
In moving around the circle a distance of 1 unit from A to
P, the angle POA is defined. The measure of this angle is
1 radian. It’s around 57.3o
Definition
One radian is the angle subtended at the centre of the
unit circle by an arc of length 1 unit.
Note: Angles formed by moving anticlockwise around the
unit circle are defined as positive; those formed by
moving clockwise are defined as negative.
Converting degrees and radians
Degrees to Radians
Radians to Degrees
𝜋
Degrees ×
180
90o=
Radians ×
180o=
270o=
2
180
𝜋
360o=
3
𝑦
𝑥
Exercise 14B: Defining Circular Functions: sine and cosine
4
The Unit Circle
The unit circle, you might remember, is the relation:
𝑥2 + 𝑦 2 = 1
It also has some very interesting properties that connect circles with the world of
trigonometry. You may need to remind yourself of the trignometry ratios:
sin(𝜃) =
cos(𝜃) =
tan(𝜃) =
𝑜𝑝𝑝
ℎ𝑦𝑝
𝑎𝑑𝑗
ℎ𝑦𝑝
𝑜𝑝𝑝
𝑎𝑑𝑗
Otherwise known by its catchy initialism SOH CAH TOA
Let’s take the unit circle as you can see below:
The x-coordinate of P(θ) is given by x = cosine θ, for θ ∈ R.
5
The y-coordinate of P(θ) is given by y = sine θ, for θ ∈ R.
These functions are usually written in an abbreviated form as follows:
x = cos θ
y = sin θ
Hence the coordinates of P(θ) are (cos θ, sin θ).
Note: Adding 2π to the angle results in a return to the same point on the unit circle. Thus
cos(2π + θ) = cos θ and sin(2π + θ) = sin θ.
𝑦
𝑥
212𝜋
6
−
49𝜋
2
𝑦
𝑥
Exercise 14C: Another circular function: tangent
One interpretation: Tangent = gradient
You may remember from previous years that the word “tangent” can be used to mean “the
gradient”. The tangent of an angle is when you do the rise over run.
tan(𝜃) =
𝑟𝑖𝑠𝑒 sin(𝜃)
=
𝑟𝑢𝑛 cos(𝜃)
This can also be proved by looking at the formula for tan(𝜃) compared to the other
trigonometric functions.
sin(𝜃) =
𝑜𝑝𝑝
ℎ𝑦𝑝
cos(𝜃) =
𝑎𝑑𝑗
ℎ𝑦𝑝
We can also prove this by considering the similar triangles
Note that tan θ is undefined when cos θ = 0.
7
tan(𝜃) =
𝑜𝑝𝑝
𝑎𝑑𝑗
The domain of tan is R \ { θ : cos θ = 0 }.
𝑦
𝑥
8
Exercise 14D: Reviewing Trigonometric Ratios
Remember the right-angled triangles:
You may need to remind yourself of the trignometry ratios:
sin(𝜃) =
𝑜𝑝𝑝
ℎ𝑦𝑝
Opposite side
𝑎𝑑𝑗
cos(𝜃) =
ℎ𝑦𝑝
𝜃
𝑜𝑝𝑝
tan(𝜃) =
𝑎𝑑𝑗
Otherwise known by its catchy initialism SOH CAH TOA
9
Adjacent side
Exercise 14E: Symmetry Properties
The coordinate axes divide the unit circle into four quadrants. The quadrants can be
numbered, anticlockwise from the positive direction of the x-axis, as shown.
10
These symmetry properties can be summarised for the signs of sin, cos and tan for
the four quadrants as follows:
1st quadrant: all are positive (A)
2nd quadrant: sin is positive (S)
3rd quadrant: tan is positive (T)
4th quadrant: cos is positive (C)
11
Negatives of Angles
𝑦
𝑥
12
13