Y2, C7 - Trigonometry & Modelling Flashcards
What are the cos addition formulae
cos(A + B) = cosAcosB - sinAsinB
cos(A - B) = cosAcosB + sinAsinB
What are the sin addition formulae
sin(A + B) = sinAcosB + cosAsinB
sin(A - B) = sinAcosB - cosAsinB
What are the tan addition formulae
tan(A + B) = (tanA + tanB) / (1 - tanAtanB)
tan(A - B) = (tanA - tanB) / (1 + tanAtanB)
What is sin(10)sin(15) - cos(10)cos(15) as a single sine, cosine or tangent
= -cos(25)
How do you create tans when given sins and cos
Divide by cos
How can we write sin15 in terms of a sin addition formula
sin(45-30) = sin45cos30 - sin45cos30
What is cos50 in terms of sin
sin40
What is sin50 in terms of cos
cos40
What is the co-function identity
sinx = cos(90 - x)
cosx = sin(90- x)
What is the sin double angle formulae
sin(2x) = 2sinxcosx
What are the cos double angle formulae
cos(2x) = cos^2(x) - sin^2(x)
= 2cos^2(x) - 1
= 1 - 2sin^2(x)
What is the tan double angle formula
tan(2x) = (2tanx) / (1 - tan^2(x))
If you cancel out a sinx or cosx in an equation, what do we know is one of the solutions
LHS = 0
How can sin(3A) be written
sin(2A + A)
Then apply double angle formula combined with addition formula
What does sinusoidal mean
Having a wavy shape
What is a harmonic identity
Expressions using sine and cosine
e.g. putting Asinx + Bcosx in the form Rsin(x + y)
Put 3sinx + 4cosx in the form R sin(x + y)
1) Expand R sin(x + y) using addition formula
2) Compare coefficients: Rcosy = 3, Rsiny = 4
3) R^2 = root(4^2 + 3^2) = 5
4) Use tan to find y, y = 53.1
5) 5sin(x + 53.1)
What is the maximum value of 12cosx + 5sinx and what is the value of x
a) 13
sinx is at a maximum when x = 1
root(144 + 25) = 13
13 x 1 = 13
12cosx + 5sinx = 13sin(x + 67.4)
b) sin(x + 67.4) = 1
x + 67.4 = 90
x = 22.6 or 382.6
What is the maximum and smallest x at max for: 2 / (10 + 3sin(x - 30)
2/7
x = 300
What does the double angle formula for cos(2 * (x/2)) look like ???
cosx = 2cos^2(x/2) -1
= 1 = 2sin^2(x/2)
Everything stays the same except the argument
