Trigonometry

Question 1

Graphing functions theta values

Answer 1

cosθ = x
sinθ = y
tanθ = sinθ/cosθ = y/x

Question 2

Graphs of trigonometric function

sin, cos and tan

notes on periodic function

notes on amplitude

notes of frequency

notes on period

Answer 2

y = sin(x), y = cos(x), y = tan(x) are three basic trigonometric graphs, they are called periodic functions as the pattern of each graph repeats itself over and over again

A periodic function repeats itself at regular intervals, a function f(x) is periodic if f(x) = f(x + a) for some value a and all values of x

The amplitude of the graph is the distance between the x-axis and the highest point on the graph

The frequency of the graph is the number of times the graph repeats itself over 2π or 360º

The distance on the x axis over which a periodic function repeats itself is called the period of the function

Question 3

Graph of y = sin(x)

at zero

max value

min value

amplitude and frequency

period

domain

range

function type

function nature

Answer 3

sin(0º) = 0, so the curve passes through the origin

Maximum value of y = sin(x) is 1

Minimum value of y = sin(x) is -1

Amplitude is 1 and frequency is 1

Period is 360º, graph repeats every 360º

Domain: x ∈ ℝ, -90º ≤ x ≤ 360º

Range: y ∈ ℝ, -1 ≤ x ≤ 1

y = sin(x) is a many to one function

sin(x) is an odd function, -sin(x) = sin(-x)
The graph has 180º rotational symmetry about the origin

Question 4

Graph of y = cos(x)

at zero

max value

min value

amplitude and frequency

period

domain

range

function type

function nature

Answer 4

sin(0º) = 1, so the curve cuts the y axis at 1

Maximum value of y = sin(x) is 1

Minimum value of y = sin(x) is -1

Amplitude is 1 and frequency is 1

Period is 360º, graph repeats every 360º

Domain: x ∈ ℝ, -π/2 ≤ x ≤ 2πº

Range: y ∈ ℝ, -1 ≤ x ≤ 1

y = cos(x) is a many to one function

cos(x) is an even function, cos(x) = cos(-x)
The graph is symmetrical about the y axis (Reflection on y-axis)

Question 5

Transformations

notation

resultant

Answer 5

y = A ___ B ( x - C) + D

___ can be sin, cos or tan

Each variable A, B, C and D can change either the shape or position of the basic graphs

Note: cos² θ = cosθ x cosθ

Question 6

Transformation of variable A

Answer 6

Variable A changes amplitude, graph is either compressed or stretches along the y-axis by the scale factor A

2 for example would make the sin graph extend to 2 and descend to -2.

For a negative factor simply swap the sign of any value inclusive of the constant for the scale factor

Question 7

Transformation of variable B

Answer 7

Variable B changes the period or frequency, graphis either compressed or stretched along the x-axis by the scale factor B, the new period for the function is 360º/B or 2π/B

2 for example would make the sin graph repeat twice in 360º and 1/2 would make the cos graph only be half complete by 2π

Question 8

Transformation of variable C

Answer 8

Variable C changes the position (movement along the x-axis, left or right) the graph is translated along the x-axis by the factor -C

To draw, draw initial graph then if C = -45 move every key point forward by 45º

Question 9

Transformation of variable D

Answer 9

Variable D changes the position (movement along the y-axis, up or down) the graph is translated along the y-axis by factor D

To draw, draw initial graph then move every point up (+) or down (-) by D

Question 10

Period properties of each graph and shape

Answer 10

Sin - sideways backwards s
Cos - downward flicked dip into a downward flicked dip
Tan - upward half u, downward half u and upward half u, downward half u

Cos and Sin is ± 2π
Tan is ± π

Question 11

Odd and even properties of each graph

Answer 11

cos(-x) = cos(x)
sin(-x) = -sin(x)
tan(-x) = -tan

Question 12

Translational properties of each graph

Answer 12

cos(x - π) = -cos(x)
cos(π - x) = -cos(x)
sin(x - π) = -sin(x)
sin(π - x) = sin(x)
tan(π - x) = -tan(x)

Question 13

Exact value of trig functions

Answer 13

30º :
sin - 1/2
cos - √3/2
tan - 1/√3

45º :
sin - 1/√2
cos - 1/√2
tan - 1

60º :
sin - √3/2
cos - 1/2
tan - √3

Question 14

Principle value/ Restricted range

Answer 14

-90º ≤ sin⁻¹(x) ≤ 90º

0º ≤ cos⁻¹(x) ≤ 180º

-90º ≤ tan⁻¹(x) ≤ 90º

Question 15

Trigonometric quadrants

Answer 15

CAST

All positive in A (0 - 90) , principle values, Q1

Sin positive in S (90 - 180), 180 - principle value, Q2

Tan positive in T (180 - 270), 180 + principle value, Q3

Cos positive in C (270 - 360) , 360 - principle values, Q4

Question 16

Sketching the graph to find solutions

Answer 16

a) roughly sketch graph and label key features
b) use calculator to find principal value
c) use principal value to find other solutions

eg.
to solve sin(x) = 0.5

draw y = sin(x) and y = 0.5

where the line and curve meet are the solutions

use a calculator to find the first (principal) x value
use quadrant rule to find any other values

Question 17

Solving trig functions by factorising

Answer 17

Let a equal to ___(x) *sin, cos or tan

factorise using quadratics to find values

sub back value into equation and solve for x

sub any other values back into equation and solve for x

exclude or reject any solutions not in given range or correct positivity/negativity

Question 18

Trigonometry Identity rules (2)

Answer 18

sin²(x) + cos²(x) = 1
tanθ = sinθ/cosθ
cosec(x) = 1/cos(x)