Trigonometry Questions

Question

1.) Find the exaqct value of tan(11π/12)

2.) A linear trig equation involving cosx has a solution of �π/6. Name three other possible solutions

3) Solve 10cosx=-7 where 0≤x≤2π

Answer 1

11π/12 is π/12 short of 2π

so tan 11π/12 = -tan π/12
I know tan 2π/12 = tan π/6 = 1/√3

tan 2π/12 = 2tanπ/12/(1 - tan^2 π/12)
let x = tan π/12 for easier typing

so
1/√3 = 2x/(1 - x^2)
2√3x = 1 - x^2
x^2 + 2√3 - 1 = 0
x = ( -2√3 ± √16)/2
tan π/12 = (4 - 2√3)/2 , since π/12 is in quad I

so tan 11π/12 = -(4-2√3)/2 = (2√3 - 4)/2

2. since cosx is positive, and x = π/6
I know the cosine is positive in I or IV
so other solutions:
2π - π/6= 5π/6
π/6 + 2π = 7π/6

3. 10 cosx = -7
cosx = -.7 , so x is in II or III
x = π - .79534 = 2.3462
x = xπ +.79534 = 3.9370

Answer 2

tan x/2 = (1-cosx)/sinx

So, if you let x=11pi/6, you can easily find sinx and cosx. Then plug them in to find tan x/2.

#2. -pi/6 11pi/6 13pi/6

#3.
cosx = -7/10
find x=arccos(7/10), then take pi±x for solutions.