PreCalculus Class Notes TEI4 Solving Trig Equations with b = 1
Finding all solutions in one period
Recall information about period and signs in quadrants
y = cos x
y = sin x
Period = 2π
Solve in [ 0, 2π )
Period = 2π
Solve in [ 0, 2π )
Solve for all solutions in one period
2
sin θ = −
2
cos θ = −
1
2
tan θ = 1
y = tan x
Period = π
⎛ π π⎞
Solve for ⎜ − , ⎟
⎝ 2 2⎠
sin θ = −1
cosθ = 2
tan θ = −
3
3
Extend to general solution: find all values
y = cos x
y = sin x
y
1
1
Graph
y = tan x
y
y
1
x
x
−4π
−7π/2
−3π
−5π/2
−2π
−3π/2
−π
−π/2
π/2
π
3π/2
2π
5π/2
3π
7π/2
4π
−4π
−7π/2
−3π
−5π/2
−2π
−3π/2
−π
−π/2
−1
2
Equation
sin θ = −
2
2
π/2
π
3π/2
2π
5π/2
3π
7π/2
4π
x
−4π
−7π/2
−3π
−5π/2
−2π
−3π/2
−π
−π/2
π/2
−1
−1
2
2
cos θ = −
1
2
π
3π/2
2π
5π/2
tan θ = 1
Soln in one
period
General
(all) soln
For the general solution, add multiples of the period to each solution: solutions + period · n,
where n is an integer
Find all solutions to the equation.
sin x = −
3
2
sec t = −1
cot θ =
3
3
3π
7π/2
4π
2cos t + 2 = 3
sin θ cos θ = 0
3sec x − 7 = −1
−2 csc x + 5 = 9
3 tan x + 5 = 4
5cot x + 5 = 0