Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
Trigonometric functions
Special angles (0, 30, 60, 90)
π
30 = 6
π
45 = 4
π
60 = 3
π
90 = 2
The general definition of an angle
An angle is a measure of the rotation of a line OF about a fixed point O.
The angle is measured from the positive x-direction. An anticlockwise rotation is taken as positive and a clockwise
rotation is taken as negative.
The Cartesian plane is divided into four quadrants and the angle 𝜃 is said to be in the quadrant where OP lies.
1
Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
Trigonometric functions
Cartesian plane
Trigonometric ratios of general angles
Signs of each trig in the four quadrants
You can memorize as ALL, SINE, TAN, COS
2
Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
Trigonometric functions
Graphs of trigonometric functions
The graph of y = sin x and y = cos x
y = sin x, 0o ≤ x ≤ 360o
y = cos x, 0o ≤ x ≤ 360o
The sine and cosine functions are called periodic functions because they repeat themselves over and over again.
The period of a periodic function is defined as the length of one repetition or cycle.
The basic sine and cosine functions repeat every 360°.
We say they have a period of 360°. (Period = 2𝜋, if working in radians.)
The amplitude of a periodic function is defined as the distance between a maximum (or minimum) point and the
principal axis.
The basic sine and cosine functions have amplitude 1.
The graph of y = tan x
y = tan x, 0o ≤ x ≤ 360o
The tangent function behaves very differently to the sine and cosine functions.
The tangent function repeats its cycle every 180° so its period is 180°.
The dashed lines at x = ±90° and x = 270° are asymptotes. The branches of the
graph get closer and closer to the asymptotes without ever reaching them.
The tangent function does not have an amplitude.
3
Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
The graphs of y = a sin bx + c, y = a cos bx + c and y = a tan bx + c
The graph of y = a sin x
4
Trigonometric functions
Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
5
Trigonometric functions
Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
6
Trigonometric functions
Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
Trigonometric equations
7
Trigonometric functions
Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
Trigonometric functions
Trigonometric identities
sin2 x + cos2 x = 1 is called a trigonometric identity because it is true for all values of x.
When proving an identity, it is usual to start with the more complicated side of the identity and prove that it
simplifies to the less complicated side.
8
Tr Naing Htet Aung (Math)
revision notes
Fb: https:/www.facebook.com/yourmathbuddy2
9
Trigonometric functions