This document provides information on trigonometric functions including definitions of sine, cosine, and tangent at common angles. It also outlines trigonometric identities, addition and double angle formulas, transformations of trig graphs, and the R-formula for expressing combinations of trig functions as a single trig function. Key concepts covered include the unit circle, quadrantal angles, amplitude, frequency, and period as they relate to trigonometric graphs.

1. AM11 Trigonometry
Angle (in radians) =
Angle (in degrees) =
(or in rad) 0° (0 rad) 30° ( rad) 45° ( rad) 60° ( rad) 90° ( rad)
sin 0 1
cos 1 0
tan 0 1 undefined
Basic angle is θ where 0 < θ < 90° if
Angle measured:
1. –θ [4th quadrant]
2. θ [1st quadrant]
3. (180° – θ) [2nd quadrant]
4. (180° + θ) [3rd quadrant]
5. (360°– θ) [4th quadrant]
6. 360n° + any of the 5
–θ 90° – θ 180° – θ 180° + θ 360° – θ 360° + θ
Angles
[4th] [1st] [2nd] [3rd] [4th] [1st]
sin – sin θ + cos θ + sin θ – sin θ – sin θ + sin θ
cos + cos θ + sin θ – cos θ – cos θ + cos θ + cos θ
tan – tan θ + cot θ – tan θ + tan θ – tan θ + tan θ

2. Equation Graph
y = sin x
For trigo graphs with
y = a sin(bx)+c & y = a cos(bx)+c,
1. Max/min value (amplitude)
= a times more
2. No of cycles (frequency)
= b times more
3. Period
= of original period
y = cos x
4. Values on the entire graph
= increased by c
Range: -1 ≤ sin x ≤1
Amplitude: (Max y – Min y)/2
For trigo graphs with
y = tan(bx) + c,
y = tan x
1. No of cycles (frequency)
= b times more
2. Period
= of original period
3. Values on the entire graph
= increased by c
Amplitude: undefined.
cosec θ = , sec θ = , tan θ =

3. 1. Simple trigonometric identities
1. tan θ =
2. cot θ =
3. sin2 θ + cos2 θ = 1
4. tan2 θ + 1 = sec2 θ
5. cot2 θ + 1 = cosec2 θ
2. Addition formulae 1. cos(x – y) = cos x cos y + sin x sin y
2. cos(x + y) = cos x cos y – sin x sin y
3. sin(x – y) = sin x cos y – cos x sin y
4. sin(x + y) = sin x cos y + cos x sin y
-
5. tan(x – y) =
6. tan(x + y) =
-
3. Double angle formulae 1. sin 2x = 2sinxcosx
2. cos 2x = cos2x – sin2x
= 2cos2x – 1
= 1 – 2sin2x
3. tan 2x =
-
4. *sin x =
5. *cos x =
6. *tan x =

4. 4. Half angle formulae 1. sin x = 2sin( )cos( )
2. cos x = cos2( ) – sin2( )
= 2cos2( ) – 1
= 1 – 2sin2( )
3. tan x =
-
5. Factor formulae -
1. sin x + sin y = 2sin( )cos( )
-
2. sin x – sin y = 2cos( )sin( )
-
3. cos x + cos y = 2cos( )cos( )
-
4. cos x – cos y = –2sin( )sin( )
6. R-formulae 1. a sin θ + b cos θ ≡ R sin(θ + α)
2. a sin θ – b cos θ ≡ R sin(θ – α)
3. a cos θ + b sin θ ≡ R cos(θ – α)
4. a cos θ + b sin θ ≡ R sin(θ + α)
R= ,α=