chapter 6 6.5 trigonometric equations
DESCRIPTION
Find all solutions of a trigonometric equation. Solve equations with multiple angles. Solve trigonometric equations quadratic in form. Use factoring to separate different functions in trigonometric equations. Use identities to solve trigonometric equations. - PowerPoint PPT PresentationTRANSCRIPT
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Chapter 66.5 Trigonometric equations
Dr .Hayk MelikyanDepartmen of Mathematics and CS
• Find all solutions of a trigonometric equation.• Solve equations with multiple angles.• Solve trigonometric equations quadratic in
form.• Use factoring to separate different functions in
trigonometric equations.• Use identities to solve trigonometric equations.• Use a calculator to solve trigonometric
equations
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Trigonometric Equations and Their Solutions
A trigonometric equation is an equation that contains a trigonometric expression with a variable, such as sin x.
The values that satisfy such an equation are its solutions. (There are trigonometric equations that have no solution.)
When an equation includes multiple angles, the period of the function plays an important role in ensuring that we do not leave out any solutions.
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• To solve an equation containing a single trigonometric function:
• Isolate the function on one side of the equation.
sinx = a (-1 ≤ a ≤ 1 )
cosx = a (-1 ≤ a ≤ 1 )
tan x = a ( for any real a )
• Solve for the variable.
Equations Involving a Single Trigonometric Function
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y = cos x
x
y
1
–1
y = 0.5
–4 2–2 4
cos x = 0.5 has infinitely many solutions for – < x <
y = cos x
x
y
1
–1
0.5
2
cos x = 0.5 has two solutions for 0 < x < 2
Trigonometric Equations
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Solve the equation: 3 sin x 2 5 sin x 1.
Solution The equation contains a single trigonometric function, sin x.
Step 1 Isolate the function on one side of the equation. We can solve for sin x by collecting all terms with sin x on the left side, and all the constant terms on the right side.
3 sin x 2 5 sin x 1 This is the given equation.
3 sin x 5 sin x 2 5 sin x 5 sin x – 1 Subtract 5 sin x from both sides.
sin x -1/2
Divide both sides by 2 and solve for sin x.
2 sin x 1 Add 2 to both sides.
2 sin x 2 1 Simplify.
Text Example
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Solve the equation: 2 cos2 x cos x 1 0, 0 x 2.
The solutions in the interval [0, 2) are /3, , and 5/3.
Solution The given equation is in quadratic form 2t2 t 1 0 with t cos x. Let us attempt to solve the equation using factoring.
2 cos2 x cos x 1 0 This is the given equation.
(2 cos x 1)(cos x 1) 0 Factor. Notice that 2t2 + t – 1 factors as (t – 1)(2t + 1).
cos x 1/2
2 cos x 1 cos x 1 Solve for cos x.
2 cos x 1 0 or cos x 1 0
Set each factor equal to 0.
Text Example
x x 2 x
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Example
cos29cos7
Solve the following equation:
Solution:
n2
5,3,
1cos
9cos9
cos29cos7
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Example
Solve the equation on the interval [0,2)
Solution:
3
3
2tan
3
7
3
6
7
62
3
3
2tan
and
and
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Example
Solve the equation on the interval [0,2)
Solution:03cos2cos2 xx
0
0
1cos3cos
01cos03cos
0)1)(cos3(cos
03cos2cos2
x
xsolutionno
xx
xx
xx
xx
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Example
Solve the equation on the interval [0,2)
Solution:
3
5,
3
2
1cos
1cos2
sincossin2
sin2sin
x
x
x
xxx
xx
xx sin2sin
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Example: Finding all Solutions of a Trigonometric Equation Solve the equation: Step 1 Isolate the function on one side of the
equation.
5sin 3sin 3. x x
5sin 3sin 3x x
5sin 3sin 3sin 3sin 3 x x x x
2sin 3x
3sin
2x
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Example: Finding all Solutions of a Trigonometric Equation (continued) Solve the equation: Step 2 Solve for the variable.
5sin 3sin 3. x x
3sin
2x
Solutions for this equation in are: 0,2
The solutions for this equation are:
2,
3 3
22 , 2
3 3n n
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Solving an Equation with a Multiple Angle
Solve the equation: tan 2 3,0 2 . x x
tan 33
Because the period is all solutions for this equation are given by
,
23
x n
6 2n
x
0n 06 2 6
x
1n3 4 2
6 2 6 6 6 3x
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Solving an Equation with a Multiple Angle
Solve the equation: tan 2 3,0 2 . x x
Because the period is all solutions for this equation are
given by
,.
6 2 nx
2n
3n
2 6 76 2 6 6 6
x
3 9 10 56 2 6 6 6 3
x
In the interval , the solutions are:2 7 5
, , , and .6 3 6 3 0,2
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Solving a Trigonometric Equation Quadratic in Form Solve the equation: 24cos 3 0, 0 2 . x x
24cos 3 0x 24cos 3x
2 3cos
4x
3 3cos
4 2x
The solutions in the interval for this equation are:
5 7 11, , , and .
6 6 6 6
0,2
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Using Factoring to Separate Different Functions
Solve the equation:sin tan sin , 0 2 . x x x xsin tan sinx x xsin tan sin 0x x x
sin (tan 1) 0x x
sin 0x
0 x x
tan 1 0x
tan 1x
5
4 4 x x
The solutions for this equation in the interval are:5
0, , , and .4 4
0,2
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Using an Identity to Solve a Trigonometric Equation
Solve the equation:cos2 sin 0, 0 2 . x x xcos2 sin 0x x
21 2sin sin 0x x 22sin sin 1 0x x
(2sin 1)(sin 1) 0x x 2sin 1 0x
2sin 1x 1
sin2
x
7 11
6 6 x x
sin 1 0x
sin 1x
x
The solutions in the
interval are7 11
, , and .6 6
0,2
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Solving Trigonometric Equations with a Calculator
Solve the equation, correct to four decimal places, for0 2 .x
tan 3.1044x 1tan (3.1044)x
1.2592x
tanx is positive in quadrants I and III
In quadrant I
In quadrant III
1.2592x
1.2592x
4.4008
The solutions for this equation are 1.2592 and 4.4008.
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Using a Calculator to Solve Trigonometric Equations Solve the equation, correct to four decimal
places, for0 2 .x
sin 0.2315x 1sin ( 0.2315)x
0.2336x
Sin x is negative in quadrants III and IV
In quadrant III
In quadrant IV 2 1.2592
6.0496
x
x
The solutions for this equation are 3.3752 and 6.0496.
0.2336
3.3752
x
x