Y2, C7 - Trigonometry & Modelling

Question 1

What are the cos addition formulae

Answer 1

cos(A + B) = cosAcosB - sinAsinB
cos(A - B) = cosAcosB + sinAsinB

Question 1

What are the sin addition formulae

Answer 1

sin(A + B) = sinAcosB + cosAsinB
sin(A - B) = sinAcosB - cosAsinB

Question 2

What are the tan addition formulae

Answer 2

tan(A + B) = (tanA + tanB) / (1 - tanAtanB)
tan(A - B) = (tanA - tanB) / (1 + tanAtanB)

Question 3

What is sin(10)sin(15) - cos(10)cos(15) as a single sine, cosine or tangent

Answer 3

= -cos(25)

Question 4

How do you create tans when given sins and cos

Answer 4

Divide by cos

Question 5

How can we write sin15 in terms of a sin addition formula

Answer 5

sin(45-30) = sin45cos30 - sin45cos30

Question 6

What is cos50 in terms of sin

Answer 6

sin40

Question 7

What is sin50 in terms of cos

Answer 7

cos40

Question 8

What is the co-function identity

Answer 8

sinx = cos(90 - x)
cosx = sin(90- x)

Question 9

What is the sin double angle formulae

Answer 9

sin(2x) = 2sinxcosx

Question 10

What are the cos double angle formulae

Answer 10

cos(2x) = cos^2(x) - sin^2(x)
= 2cos^2(x) - 1
= 1 - 2sin^2(x)

Question 11

What is the tan double angle formula

Answer 11

tan(2x) = (2tanx) / (1 - tan^2(x))

Question 12

If you cancel out a sinx or cosx in an equation, what do we know is one of the solutions

Answer 12

LHS = 0

Question 13

How can sin(3A) be written

Answer 13

sin(2A + A)
Then apply double angle formula combined with addition formula

Question 14

What does sinusoidal mean

Answer 14

Having a wavy shape

Question 15

What is a harmonic identity

Answer 15

Expressions using sine and cosine
e.g. putting Asinx + Bcosx in the form Rsin(x + y)

Question 16

Put 3sinx + 4cosx in the form R sin(x + y)

Answer 16

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)

Question 17

What is the maximum value of 12cosx + 5sinx and what is the value of x

Answer 17

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

Question 18

What is the maximum and smallest x at max for: 2 / (10 + 3sin(x - 30)

Answer 18

2/7
x = 300

Question 19

What does the double angle formula for cos(2 * (x/2)) look like ???

Answer 19

cosx = 2cos^2(x/2) -1
= 1 = 2sin^2(x/2)
Everything stays the same except the argument