Thursday: 1 - 2 p.m.
Room: MS313
Solving Trigonometric Equations
I. Inverse Trigonometric Functions
In order to construct the inverse trigonometric functions, we must first restrict the angular
domain of the trigonometric function to ensure that it is one-to-one.
Definition
i) Inverse Sine:
sin-1x = θ if sin θ = x and θ in [-90⁰, 90⁰]
This means that the output of the inverse sine function is equal to the angle whose value is x
with the resulting angle lying in the range [-90⁰, 90⁰] .
Note: x must be in [-1,1]
Ex: Find the following values without a calculator:
a) sin-11
b) sin-1(
c) sin-1(- )
)
d) sin-10
e) sin-12
f) sin-1 (0.222) (calculator)
i) Inverse Cosine:
cos-1x = θ if cosθ = x and θ in [0⁰, 180⁰]
This means that the output of the inverse cosine function is equal to the angle whose value is x
with the resulting angle lying in the range [0⁰, 180⁰] .
Note: x must be in [-1,1]
Ex: Find the following values without a calculator:
a) cos-11
b) cos -1(
) c) cos -1(- )
d) cos -10 e) cos -12
f) cos-1 (-0.222) (calculator)
tan-1x = θ if tan θ = x and θ in (-90⁰, 90⁰)
iii) Inverse Tangent:
This means that the output of the inverse tangent function is equal to the angle whose value is
x with the resulting angle lying in the range (-90⁰, 90⁰).
Note: x can be any real number
Ex: Find the following values without a calculator:
a) tan-11
b) tan-1(
)
c) tan-1(-
)
d) tan-10
e) tan-12 (calculator)
II. Solving Trigonometric Equations that are linear in a trigonometric function (ie. no sin2x etc.)
The process is similar to that of solving a linear equation in x but this time we solve for the
trigonometric function of x (or some other variable). Then we must find all values of the angle
for which the trigonometric function will take on that value.
Ex: Solve for all values of x in [0, 2π].
Ex: Solve for all values of x in [0, 2π].
2sinx + 3 = 3
cosx = -1
Ex: Solve for all values of x in[0, 2π].
Ex: Solve for all values of x in [0, 2π].
cosx = 0
tanx = 0
Note: the values that we are seeing on the right side of the equation correspond to quadrantal
angles.
In these cases, using the unit circle approach works well to find all solutions.
When the numeric value on the right does not correspond to a quadrantal angle we must us a
two step process.
Ex: Solve sinx =
for all value of x in [0, 360⁰].
Step 1: Find the reference angle for which sine is
Step 2: To find all solutions for which sine is +
. This angle is x' = 30⁰
we must determine the 2 quadrants in which
sine is positive. These are QI and QII so our two solutions are given as follows are 30⁰ and
150⁰. Can you explain how the second solution was obtained?
Ex: Same problem but with no limited interval in which the solutions must lie.
Solve: sinx =
When no interval is specified it means that you must provide all solutions! Since the
trigonometric functions are periodic you can use the period to describe all solutions as follows:
x=
, where k is any integer.
Can you list 6 different solutions?
Ex: Solve: cosx =
for all value of x in [0, 2π].
Ex: Solve: cosx =
(meaning all values)
Ex: Solve: tanx =
for all value of x in [0, 2π]. Ex: Solve: tanx =
(meaning all values)
Ex: Solve: 2cosx -
=0
Ex: Solve: 4sinx - 4 = -2
For the problems below you will need a calculator to find the reference angle.
Ex: Solve: sinx = 0.444 for all value of x in [0, 2π].
Ex: Solve: cosx = - 0.228
Ex: Solve: tanx = 8.888
III. Multiple Angle Equations
(If the power on the trigonometric function is 1, solve without applying an identity)
Ex: Solve: sin(4x) = Let θ = 4x: sin(θ) = ›>
›> θ' = 30⁰
4x = θ =
›>
x=
›>
x=
›>
θ=
, where k is any integer.
, where k is any integer.
, where k is any integer.
, where k is any integer.
Ex: Solve cos(2x) =
Ex: Solve: sin(3x) = 0.428
Ex: Solve: tan(2x) = -1
IV. Quadratic Equations
Ex: Solve: 2sin2x - sinx = 1
Ex: Solve: 2sinxcosx - sinx = 0
(Considered Quadratic since we have a product of 2 functions)
Ex: Solve: cos2x - 1 = 0
Ex: Solve: tan2x - 4 = 0