Download Solving Trigonometric Equations

Warm UP
Grab a Trig Cheat Sheet from the back table
Solve these using your unit circle:
1. Arcsin(1/2) =
2. arccos(0) =
Now try to solve these
3. sin(x) = ½
4. cos(x) = 0
Math
IV
Lesson
37
Essential Question:
How can I use inverses to solve trigonometric
equations? What are the different mentods to
solve trigonometric equations?
Standards:
MM4A6. Students will solve trigonometric equations both
graphically and algebraically.
a. Solve trigonometric equations over a variety of domains,
using technology as appropriate.
MM4A8. Students will investigate and use inverse sine,
inverse cosine, and inverse
tangent functions.
a. Find values of the above functions using technology as
appropriate.
Introduction to solving
Trigonometric equations
Your preliminary goal in solving a trigonometric equation is
to isolate the trigonometric function in the equation.
To solve a trigonometric equation, use standard algebraic
techniques such as collecting like terms and factoring.
For example, to solve the equation 2 sin x = 1, divide each
side by 2 to obtain
How many solutions are there to trig
equations
….. well how many times can you go around
the unit circle?
However, We will be talking about solving trig
equations on the interval from [0,2ㅠ]
Methods to solving equations
Finding the angles that make the statement true.
Several methods for solving these types of equations
that are similar to methods used with polynomials.
–Combining like terms
–Taking square roots
–Factoring
Solving
Solving Trigonometric
Equations
Trigonometric
Equations
Solve:
2cos x 1  0
Step 1: Isolate cos x using algebraic skills.
2cos x  1
cos x  1
2
Step 2: Use the inverse function to assist . Or use your Unit Circle to find the value of x
Solve:
Step 1:
tan 2 x  1  0
tan 2 x  1
tan 2 x   1
tan x  1
Step 2: Solve using your Unit Circle
Solve:
Step 1:
cot x cos 2 x  2 cot x
cot x cos 2 x  2 cot x  0
cot x  cos 2 x  2   0
cot x  0 or cos 2 x  2  0
cos 2 x  2
cos2 x   2
cos x   2
Step 2: Solve for x
Try these:
1.
tan x  1  0
2.
sec 2 x  4  0
3.
3 tan 3 x  tan x
Solution
Solve:
2 sin 2 x  sin x  1  0
 2sin x  1 sin x 1  0
Factor the quadratic equation.
2 sin x  1  0
sin x  1  0
Set each factor equal to zero.
sin x  1
Solve for sin x
sin x  
1
2
or
Determine the solution(s).
remember cos(Ө) = x , sin(Ө) = y and
tan(Ө) = y/x
Using the unit circle to get exact values, solve
for x, [0, 2π):
1.cos x = ½
2.sin x = 0
3.tan x = ±√3
4.cos x = -1
Many times you will have to manipulate the
equation to solve
1.Combining like terms
2.Using identities to simplify
3.Factoring
Try these:
Solve for x:
0 = 4x – 2
0 = 4sin x – 2
Now try these:
Solve for x:
2
0 = 2x + 3x + 1
2
0 = 2cos x + 3cos x + 1
4sin2 x – 3 = 0
sin x – cos x sin x = 0
sin x (sec2 x + 1) = 0
Solve:
Solution:
cos 3 x 
1
2
No algebraic work needs to be done because cosine is already by itself.
Remember, 3x refers to an angle and one cannot divide by 3 because it
is cos 3x which equals ½.
Since 3x refers to an angle, find the angles whose cosine value is ½.
Now divide by 3 because it is angle equaling angle.
Notice the solutions do not exceed 2. Therefore,
more solutions may exist.
Try these:
sin 2 x  
cos
3
2
x
2

2
2
A Few Rules:
1.Look for values on the unit circle
2.When taking the square root, don’t forget the + and –
3.Never divide by a variable , move to the other side of the
equation and factor!
Given 3tan3 x = tan x
Subtract tan from both sides 3tan3 x – tan x = 0
Factor tan x(3tan2 x – 1) = 0
Solve: tan x = 0 or 3tan2 x – 1 = 0
tan2 x = 1/3