5.3 S OLVING T RIG EQUATIONS. S OLVING T RIG E QUATIONS Solve the following equation for x: Sin x =...
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Transcript of 5.3 S OLVING T RIG EQUATIONS. S OLVING T RIG E QUATIONS Solve the following equation for x: Sin x =...
SOLVING TRIG EQUATIONS
In this section, we will be solving various types of trig equations
You will need to use all the procedures learned last year in Algebra II
All of your answers should be angles.
Note the difference between finding all solutions and finding all solutions in the domain [0, 2π)
SOLVING TRIG EQUATIONS
Guidelines to solving trig equations:
1) Isolate the trig function
2) Find the reference angle
3) Put the reference angle in the proper quadrant(s)
4) Create a formula for all possible answers (if necessary)
SOLVING TRIG EQUATIONS
1- 2 Cos x = 0
1) Isolate the trig function
1- 2 Cos x = 0 + 2 Cos x = + 2 Cos x
1= 2 Cos x 2 2
Cos x = ½
SOLVING TRIG EQUATIONS
Cos x = ½
2) Find the reference angle
x = 3
3) Put the reference angle in the proper quadrant(s)
I = 3
IV =
3
5
SOLVING TRIG EQUATIONS
Find all solutions to the following equation:
Sin x + 1 = - Sin x+ Sin x + Sin x
→ 2 Sin x + 1 = 0- 1 - 1
→ 2 Sin x = -1→ Sin x = - ½
SOLVING TRIG EQUATIONS
Find the solutions in the interval [0, 2π) for the following equation:
Tan²x – 3 = 0
Tan²x = 3Tan x = 3
SOLVING TRIG EQUATIONS
Tan x = 3
Ref. Angle: Quad.:3
IV III, II, I,
I:3
II:
3
2 III:3
4IV:
3
5
x = ,3
,
3
2,
3
43
5
SOLVING TRIG EQUATIONS
Solve the following equations for all real values of x.
a) Sin x + = - Sin x
b) 3Tan² x – 1 = 0
c) Cot x Cos² x = 2 Cot x
2
SOLVING TRIG EQUATIONS
Find all solutions to the following equation:
Sin x + = - Sin x
2 Sin x = -
Sin x = -
2
2
2
2
x = 4
5
x = 4
7
2n
2n
SOLVING TRIG EQUATIONS
3Tan² x – 1 = 0
Tan² x = 3
1
Tan x = 3
1
x = 6
6
5x =
6
7x =
6
11x =
2n
2n
2n
2n
SOLVING TRIG EQUATIONS
Cot x Cos² x = 2 Cot x
Cot x Cos² x – 2 Cot x = 0
Cot x (Cos² x – 2) = 0
Cot x = 0 Cos² x – 2 = 0
Cos² x – 2 = 0 Cos x = 2
Cos x = 0 x =
2
x = 2
3
2n
2n No Solution
SOLVING TRIG EQUATIONS
Find all solutions to the following equation.
4 Tan²x – 4 = 0 Tan²x = 1 Tan x = ±1 Ref. Angle =
4
x = 4
4
3x =
n
n
SOLVING TRIG EQUATIONS
Equations of the Quadratic Type
Many trig equations are of the quadratic type:
2Sin²x – Sin x – 1 = 0 2Cos²x + 3Sin x – 3 = 0
To solve such equations, factor the quadratic or, if that is not possible, use the quadratic formula
SOLVING TRIG EQUATIONS
Solve the following on the interval [0, 2π)
2Cos²x + Cos x – 1 = 0
If possible, factor the equation into two binomials.
2x² + x - 1
(2Cos x – 1) (Cos x + 1) = 0
Now set each factor equal to zero
SOLVING TRIG EQUATIONS
2Cos x – 1 = 0 Cos x + 1 = 0
Cos x = ½
Ref. Angle: 3
Quad: I, IV
x = ,3
3
5
Cos x = -1
x =
SOLVING TRIG EQUATIONS
Solve the following on the interval [0, 2π)
2Sin²x - Sin x – 1 = 0
(2Sin x + 1) (Sin x - 1) = 0
SOLVING TRIG EQUATIONS
2Sin x + 1 = 0 Sin x - 1 = 0
Sin x = - ½
Ref. Angle: 6
Quad: III, IV
x = ,6
76
11
Sin x = 1
x = 2
SOLVING TRIG EQUATIONS
Solve the following on the interval [0, 2π)
2Cos²x + 3Sin x – 3 = 0
Convert all expressions to one trig function
2 (1 – Sin²x) + 3Sin x – 3 = 02 – 2Sin²x + 3Sin x – 3 = 0
0 = 2Sin²x – 3Sin x + 1
SOLVING TRIG EQUATIONS
2Sin x - 1 = 0 Sin x - 1 = 0
Sin x = ½
Ref. Angle: 6
Quad: I, II
x = ,6
6
5
Sin x = 1
x = 2
0 = 2Sin²x – 3Sin x + 1 0 = (2Sin x – 1) (Sin x – 1)
SOLVING TRIG EQUATIONS
Solve the following on the interval [0, 2π)
2Sin²x + 3Cos x – 3 = 0
Convert all expressions to one trig function
2 (1 – Cos²x) + 3Cos x – 3 = 02 – 2Cos²x + 3Cos x – 3 = 0
0 = 2Cos²x – 3Cos x + 1
SOLVING TRIG EQUATIONS
2Cos x - 1 = 0 Cos x - 1 = 0
Cos x = ½
Ref. Angle: 3
Quad: I, IV
x = ,3
3
5
Cos x = 1
x = 0
0 = 2Cos²x – 3Cos x + 1 0 = (2Cos x – 1) (Cos x – 1)
SOLVING TRIG EQUATIONS
The last type of quadratic equation would be a problem such as:
Sec x + 1 = Tan x
What do these two trig functions have in common?
When you have two trig functions that are related
through a Pythagorean Identity, you can square
both sides.
( )² ²
SOLVING TRIG EQUATIONS
(Sec x + 1)² = Tan²x
Sec²x + 2Sec x + 1 = Sec²x - 1
2 Sec x + 1 = -1 Sec x = -1
Cos x = -1
x = When you have a problem that requires you to square both sides, you must check your answer when you are done!
SOLVING TRIG EQUATIONS
Cos x + 1 = Sin x
Cos²x + 2Cos x + 1 = 1 – Cos² x
2Cos² x + 2 Cos x = 0Cos x (2 Cos x + 2) = 0
Cos x = 0
x = ,2
(Cos x + 1)² = Sin² x
Cos x = - 1
2
3x =
SOLVING TRIG EQUATIONS
Cos x + 1 = Sin x
x = ,2
,
2
3 2
Sin 1 2
Cos
1 1 0
2
3Sin 1
2
3 Cos
1- 1 0
Sin 1 Cos
0 1 1-
SOLVING TRIG EQUATIONS
Equations involving multiply angles
Solve the equation for the angle as your normally would
Then divide by the leading coefficient
SOLVING TRIG EQUATIONS
Solve the following trig equation for all values of x.
2Sin 2x + 1 = 02Sin 2x = -1
Sin 2x = - ½
2x = 2n
6
7 2x =
2n 6
11
x = n
12
7 x =
n 12
11
SOLVING TRIG EQUATIONS
0 3 2
xTan 3
1- 2
xTan
2
x n
4
3
2
x n
4
7
2
x 2n
2
3
2
x 2n
2
7
Redundant Answer
SOLVING TRIG EQUATIONS
Solve the following equations for all values of x.
a) 2Cos 3x – 1 = 0
b) Cot (x/2) + 1 = 0
SOLVING TRIG EQUATIONS
2Cos 3x - 1 = 02Cos 3x = 1
Cos 3x = ½
3x = 2n
3 3x =
2n 3
5
x = 3
2n
9
x =
3
2n
9
5
SOLVING TRIG EQUATIONS
Topics covered in this section:
Solving basic trig equations Finding solutions in [0, 2π) Find all solutions
Solving quadratic equations Squaring both sides and solving Solving multiple angle equations Using inverse functions to generate answers
SOLVING TRIG EQUATIONS
Find all solutions to the following equation:
Sec²x – 3Sec x – 10 = 0(Sec x + 2) (Sec x – 5) = 0
Sec x + 2 = 0 Sec x – 5 = 0Sec x = -2Cos x = - ½
x = 2n 3
2
2n 3
4
Sec x = 5Cos x =
5
1
x =
5
1Cos 1-
SOLVING TRIG EQUATIONS
One of the following equations has solutions and the other two do not. Which equations do not have solutions.
a) Sin²x – 5Sin x + 6 = 0b) Sin²x – 4Sin x + 6 = 0c) Sin²x – 5Sin x – 6 = 0
Find conditions involving constants b and c that
will guarantee the equation Sin²x + bSin x + c = 0
has at least one solution.
SOLVING TRIG FUNCTIONS
Find all solutions of the following equation in the interval [0, 2π)
Sec²x – 2 Tan x = 4
1 + Tan²x – 2Tan x – 4 = 0
Tan²x – 2Tan x – 3 = 0
(Tan x + 1) (Tan x – 3) = 0
Tan x = -1 Tan x = 3