chapter 6 circular functions - my maths

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CIRCULAR FUNCTIONS

6A Trigonometric ratio revision

EXAMPLE

Find the value of in each of the following triangles.

(a)

(b)

(c)

CIRCULAR FUNCTIONS of 27

6B The unit circle

The unit circle is a circle with center 0, 0 and a radius of 1 unit.

At point ,

the ‐coordinate is ____________.

the ‐coordinate is ____________.

The “tangent line” on a unit circle is the line that is a tangent to the circle at the point 1, 0 .

tan is the height at which a line along an angle hits the tangent line.

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Quadrants

The coordinate axes divide the unit circle into four quadrants as shown in the diagram below.

EXAMPLE

If is any angle, find the quadrant where:

(a) sin ° 0 and tan ° 0

(b) sin ° 0 and cos ° 0.


Exact values

Angles of sin, cos or tan are taken from the positive ‐axis, traversing in an anticlockwise direction.

Consider two basic right angled triangles with common angles 30°, 45° and 60°.

Angle ( °) sin cos tan

0 0 1 0

30 1

2

√3

2

1

√3

45

1

√2

1

√2

1

60 √3

2

1

2 √3

90 1 0 undefined


EXAMPLE

Without using a calculator, find:

(a) sin 90°

(b) cos 180°

(c) tan 270°

EXAMPLE

Without using a calculator, state the exact value of:

(a) sin 45°

(b) cos 60°

(c) tan 30°


6C Radians

An alternative unit for angle measurement is the radian (c), on the calculator as (r).

Radians are derived from the circumference of the unit circle:

In a unit circle,

Conversions

Degrees to radians Radians to degrees

the circumference


EXAMPLE

Convert the following angles into exact radians.

(a) 120°

(b) 270°

EXAMPLE

Convert the following angles to degrees.

(a)

(b) 0.8

EXAMPLE

Use a calculator to convert:

(a) 57.2° to radians, correct to 3 decimal

places.

(b) 2.75c to the nearest tenth of a degree.


Exact values and radians

Since 180°,

30°, 45°, 60° and 90°.

So the table of exact values can be written in terms of radian measurements:

Angle ( ) sin cos tan

0 0 1 0

6

1

2

√3

2

1

√3

4

1

√2

1

√2

1

3

√3

2

1

2 √3

2 1 0 undefined


6D Symmetry

The unit circle can be divided into symmetrical sections, as shown in the diagram below.

Relationships between the circular functions (sine, cosine and tangent) can be established, based

on these symmetrical properties.

For simplicity, assume is an acute angle, although the following properties hold for any .

2nd quadrant

From symmetry:

sin

cos

tan

1st quadrant

As already seen:

sin

cos

tan

3rd quadrant

From symmetry:

sin

cos

tan

4th quadrant

From symmetry:

sin 2

cos 2

tan 2

Note: An angle measurement is assumed to be in radians unless the degree symbol is given.


Provided an angle is expressed as or 2 , the trigonometric

function remains the same, only the sign ( or ) may change.

Use this diagram to determine the sign.

EXAMPLE

(a) If sin 0.93, find sin 180° .

(b) If cos 0.44, find cos 360° .

(c) If tan 1.72, find tan .

(d) If cos 0.83, find cos .


EXAMPLE

Without using a calculator, find the exact value of each of the following.

(a) tan 150°

(b) sin 330°

(c) cos

(d) tan


EXAMPLE

If sin 0.924, evaluate each of the following:

(a) sin

(b) sin


6E Identities

Pythagorean identity

sin cos 1

Tangent identity

tansin

cos

Complementary functions

sin 90° cos °

cos 90° sin °

or equivalently

sin2

cos

cos2

sin

EXAMPLE

If sin 0.4 and 0° 90°, find, correct to 3 decimal places:

(a) cos

(b) tan


EXAMPLE

Find all possible values of sin if cos 0.75.

EXAMPLE

If 0° ° 90° and cos ° , find the exact values of:

(a) sin °

(b) cos 90° °

(c) tan °

(d) sin 180° °


6F Sine and cosine graphs

The graph of

The graph of

In general:

sin

cos

where

the amplitude is

the period is

the median is

the maximum is

the minimum is

Note: The amplitude is always positive.

If 0, then


EXAMPLE

State i. the period and ii. the amplitude of each of the following functions.

(a)

(b) 1.5 sin 4

EXAMPLE

Sketch the graphs of the following functions and state i. the period and ii. the amplitude of each.

(a) 4 cos , for ∈ 0, 12

(b) sin 4 , for ∈ 0, 2


EXAMPLE:

(a) Sketch the graph of cos 2 3, ∈ 0, 2 , and state i. the period ii. the amplitude and

iii. the maximum and minimum values.

(b) Sketch the graph of cos 2 3, ∈ 0, 2 , using a CAS calculator.

Ensure your calculator is in radian mode. On a Graphs &

Geometry page, complete the entry line as:

1 cos 2 3|0 2

Then press enter.

Note: The Window Settings should be adjusted.

EXAMPLE

Sketch the graph of the following function:

: 2,4 → , 2cos2


6G Tangent graphs

The graph of

EXAMPLE

For each function below, state i. the period and ii. the equation of the two asymptotes closest to

the ‐axis.

(a) tan 4

(b) 3 tan 3


EXAMPLE

Sketch the graph of the function 3 tan over ∈ 2, 4 :

(a) without a CAS calculator

(b) with a CAS calculator

Ensure your calculator is in radian mode.

On a Graphs & Geometry page, complete the entry line as:

1 3 tan2

| 2 4

Then press enter.

Note: The Window Settings should be adjusted.


6H Solving trigonometric equations

To find the solution to the equation sin√ where ∈ 0, 4 , we can consider the graph of

sin .

By drawing a horizontal line through √, it can be seen that there are four solutions in the

domain 0, 4 .

The solution for 0 , that is, in the first quadrant, is (from our knowledge of exact values).

Note: For inexact solutions in the first quadrant, use a calculator.

The sine function is also positive in the second quadrant.

Using symmetry, the next solution is

Since the graphs are periodic, any further solutions are found by adding (or subtracting) the period

(in this case 2 ) to (or from) each of the first two solutions.

For example, two further solutions are:

42 and

3

42

and

Therefore, four solutions in the specified domain are:

However, if a domain is not specified, there are an infinite number of solutions as multiples of 2

can be added (or subtracted) indefinitely to (or from) and .

In this situation a general solution is obtained where the solutions are in terms of a parameter,

where is an integer, i.e. ∈ .


In general:

If sin , then 2 arcsin and 2 1 arcsin ;

where ∈ 1, 1 and ∈ .

If cos , then 2 arccos ; where ∈ 1, 1 and ∈ .

If tan , then arctan ; where ∈ and ∈ .

When solving trigonometric equations, the following need to be determined:

the first quadrant angle, irrespective of the sign.

the two quadrants in which the given function is positive or negative.

the two solutions between 0 and 2 , using the appropriate symmetry property.

If more solutions are required:

Repeatedly add (or subtract) the period to the two solutions as many times as required,

noting solutions after each addition or subtraction.

Stop when all solutions within the specified domain are found.

If no domain is given, a general solution is required.

EXAMPLE

Find to the nearest tenth of a degree if cos ° 0.58, given that ∈ 0°, 360° .


EXAMPLE

Find the value of in each of the following equations if ∈ 0, 4 . Give answers correct to 3

decimal places, unless exact answers may be found.

(a) sin 0.3

(b) cos


(c) Solve parts (a) & (b) using a CAS calculator.

Ensure your calculator is in radian mode.

EXAMPLE

Find solutions to 2 sin 0.984 over the domain 0, 2 .


EXAMPLE

Find the general solution of the following equations. Hence, find all the solutions for

2 2 for each equation.

(a) cos√

Note the answer from step 2 could be further simplified by

combining the two terms. A CAS calculator will give the answer in

this form.

(b) 2 sin 1 0


(c) tan √3


6I Applications

Many situations arise in science and nature where relationships between two variables exhibit

periodic behavior. Tide heights, sound waves, biorhythms and ovulation cycles are examples.

In these situations trigonometric functions can be used to model the behavior of the variables. The

independent variable is often a measurement such as time.

When modelling with trigonometric functions you should work in radians unless otherwise

instructed.

EXAMPLE

E. coli is a type of bacterium. Its concentration, parts per million (ppm), at a particular beach over

a 12 hour period hours after 6 am, is described by the function:

0.05 sin12

0.1

(a) Find the i. maximum and ii. minimum E. coli levels at this beach.

(b) What is the level at 3 pm?


(c) How long is the level above 0.125 ppm during the first 12 hours after 6 am?

Sketch the graph first to see the solutions.

chapter 6 circular functions - my maths

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