Trigonometry Flashcards
Angle measurement
Angles can be measured in radians or degrees but radians are preferred as it makes dealing with trigonometric functions easier
One radian is the measure of the angle subtended at the centre of a circle with a radius of one unit, by an arc length one unit
π radian = 180º
Special triangles
1)
Triangle with angles 45º and corresponding side lengths 1, with the hypotenuse √2
2)
Triangle with angles 60º and 30º and corresponding sides √3 and 1 respectively, with the hypotenuse 2
Graphs of trig functions
y = sin(x)
upside down sideways “s”
y = cos(x)
curvy “u”
y = tan(x)
half “u”, upside down half “u” into half “u”, upside down half “u”
Period properties
cos(x ± 2π) = cos(x)
sin(x ± 2π) = sin(x)
tan(x ± π) = tan(x)
Odd and even properties
cos(-x) = cos(x)
sin(-x) = -sin(x)
tan(-x) = -tan(x)
Translation properties
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)
Reciprocal trig functions
The reciprocals of the 3 basic trig functions are;
sin(x) → cosecant(x) [cosec(x)]
cos(x) → secant(x) [sec(x)]
tan(x) → cotangent(x) [cot(x)]
cosec(x) = 1 / sin(x)
sec(x) = 1 / cos(x)
cot(x) = 1 / tan(x)
Note:
cot(x) = 1/tan(x), except where tan(x) = 0
if where tan(x) is 0 cot(x) is undefined and where tan(x) is undefined cot(x) is 0
Graphs of reciprocal trig functions
y = cosec(x)
Domain: x ∈ real numbers,
x ≠ -180º, 180º, 360º, …
Rangle: y ∈ real numbers, y ≤ -1 or y ≥ 1
The period of the curve is 360º (2π) and it is an odd function so that it has a half turn rotational symmetry about the origin (0,0).
y = sec(x)
Domain: x ∈ real numbers,
x ≠ -90º, 90º, 270º, …
Rangle: y ∈ real numbers, y ≤ -1 or y ≥ 1
The period of the curve is 360º (2π) and it is an even function so that it is symmetrical about the y-axis.
y = cot(x)
Domain: x ∈ real numbers,
x ≠ -180º, 0º, 180º, 360º, …
Rangle: y ∈ real numbers
The period of the curve is 180º (π) and it is an odd function so that it has a half turn rotational symmetry about the origin (0,0).
Trig identities
An identity is an equation which is true for all values of the unknown, whereas an equation is usually true for only one or some values
sin²(x) + cos²(x) ≡ 1
1 + cot²(x) ≡ cosec²(x)
tan²(x) + 1 ≡ sec²(x)
Solving trig equations
Sketching the graph
1. Sketch the graph of the trig functions accurately, label all the key features
2. Use the calculator to find the principal value
3. Use the principal value to find other solutions
Trig quadrants
CAST diagram
functions all positive in A and their respective letters
sin(x) = k
cos(x) = k
tan(x) = k
- k will be either positive or negative, this corresponds with the 2 quadrants containing the solutions
- Use the calculator, take the inverse of the function and the k value to obtain the principal angle (Q1)
- Q1 (done), Q2 (180º– Q1), Q3 (180º + Q1), Q4 (360º – Q1)
Compound angle formulae
sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
sin(A – B) = sin(A) cos(B) – cos(A) sin(B)
cos(A + B) = cos(A) cos(B) – sin(A) sin(B)
cos(A – B) = cos(A) cos(B) + sin(A) sin(B)
tan(A + B) = [ tan(A) + tan(B) ] / 1– tan(A) tan(B)
tan(A – B) = [ tan(A) – tan(B) ] / 1+ tan(A) tan(B)
Double angle formulae
A useful simplification of the compound angle formulae is when A and B are equal, A + B = A + A = 2A
sin(2A) ≠ 2sin(A)
sin(2A) = sin(A)cos(A)
cos(2A) = 1 – 2sin²(A)
tan(2A) = [ 2tan(A) ] / 1 – tan²(A)
Note: the 3 versions of the cosine double angle formula are obtained using the identity sin²(x) + cos²(x) ≡ 1
sin(3x) = sin(2x + x)
The form a sin(x) + b cos(x)
Example:
y = 3 sin(x) + 2 cos(x)
done in form specified ie. R sin(x + a)
Use compound angle formula to equate,
let x equal values to cancel out either value eg. 0º or 90º
2 equations derived substituted into special triangle to find x
Hence then solve for y
To find maximum and minimum use + and – R values, find corresponding x values and use equation to solve
