Trigonometry Flashcards
Graphing functions theta values
cosθ = x sinθ = y tanθ = sinθ/cosθ = y/x
Graphs of trigonometric function
sin, cos and tan
notes on periodic function
notes on amplitude
notes of frequency
notes on period
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
Graph of y = sin(x)
at zero
max value
min value
amplitude and frequency
period
domain
range
function type
function nature
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
Graph of y = cos(x)
at zero
max value
min value
amplitude and frequency
period
domain
range
function type
function nature
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)
Transformations
notation
resultant
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
Transformation of variable A
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
Transformation of variable B
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
Transformation of variable C
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
Transformation of variable D
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
Period properties of each graph and shape
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
Odd and even properties of each graph
cos(-x) = cos(x) sin(-x) = -sin(x) tan(-x) = -tan
Translational properties of each graph
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)
Exact value of trig functions
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
Principle value/ Restricted range
-90º ≤ sin⁻¹(x) ≤ 90º
0º ≤ cos⁻¹(x) ≤ 180º
-90º ≤ tan⁻¹(x) ≤ 90º
Trigonometric quadrants
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
Sketching the graph to find solutions
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
Solving trig functions by factorising
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
Trigonometry Identity rules (2)
sin²(x) + cos²(x) = 1
tanθ = sinθ/cosθ
cosec(x) = 1/cos(x)
