7. Trigonometric Modelling

Question 1

sin(A+B)

Answer 1

sinA cosB + cosA sinB

Question 2

sin(A-B)

Answer 2

sinA cosB - cosA sinB

Question 3

cos(A+B)

Answer 3

cosA cosB - sinA sinB

Question 4

cos(A-B)

Answer 4

cosA cosB + sinA sinB

Question 5

tan(A+B)

Answer 5

tanA + tanB
———————
1 - tanA tanB

Question 6

tan(A-B)

Answer 6

tanA - tanB
———————
1 + tanA tanB

Question 7

Proving cos(-B)

Answer 7

Substitute B = -B into cos(A+B)

Question 8

Proving tan(A+B)

Answer 8

Write as sin(A+B)/cos(A+B) and divide numerator and denominator by cosA cosB

Question 9

Using the addition formulae to show sin/cos/tan x = …

Answer 9

Use 0,30,45,60 or 90 in the addition formulae

Question 10

sin2A

Answer 10

2sinAcosA

Question 11

cos2A

Answer 11

cos^2(A) - sin^2(A)
2cos^2(A) - 1
1 - 2sin^2(A)

Question 12

tan2A

Answer 12

1-tan^2(A)

Question 13

Double angle formulae derivation

Answer 13

Substitute A = B into the addition formulae

Question 14

b sinx + c cosx to R sin/cos(x+a)

Answer 14

1. Expand the RHS using the addition formulae
2. Set the coefficients of the LHS and from your expansion equal
3. Square Rsina and Rcosa and root the sum for R
4. Use tan a = Rsina/Rcosa to find a
5. Substitute into the formula

Question 15

Maximising and minimising with R formulae

Answer 15

The maximum is R, the minimum -T

Find what x must be so the expression in the bracket equals 1 or -1 depending on what is needed