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Solving Trigonometric Equations Quadrant I Quadrant II Quadrant IIIQuadrant IV Cosine All Sine Tangent Exact Values of Special Angles
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Sect. 5-3
What is SOLVING a trig equation?
• It means finding the values of x that will make the equation true. (Just as we did with algebraic equations!)
• Until now, we have worked with identities, equations that are true for ALL values of x. Now we’ll be solving equations that are true only for specific values of x.
Solving Trigonometric EquationsQuadrant IQuadrant II
Quadrant III Quadrant IV
Cosine
AllSine
Tangent
1800 - q1800 + q 3600 - q
q
12
12
32
13
32
3
12 1
12
Exact Values of Special Angles
Solving Trigonometric Equations
sinq 3
2
q 3
q 3
,23
1) tanq 1
q 4
q 4
,54
2)
Solve for q if 0 ≤ q < 2.
General Solutions22 , 2
3 3..., 2, 1,0,1,2
n n
n
q + +
- -
ReferenceAngle
ReferenceAngle
General Solutions
q 4+ 2n, 5
4+ 2n
n I
sin2 q 12
q 4
q 4
,34
,54
,74
3)
sinq 12
sinq 2
2
Solving Trigonometric Equations
Solve for q if 0 ≤ q < 2.
ReferenceAngle
sinq 12
4) 2tan 1 0x - 2tan 1x
2tan 1x tan 1x
3 5 7, , ,4 4 4 4
x
Q1 QII QIII QIV
5)
Solving Trigonometric Equations
Solve for q if 0 ≤ q < 2.
3 cscq - 2 03 cscq 2
cscq 23
q
3
, 23
6) 4 cosq+ 3 2 cosq + 22 cosq - 1
cosq -12
q
23
, 43
1. Try to get equations in terms of one trig function by using identities.
6. Try to get trig functions of the same angle. If one term is cos2q and another is cosq for example, use the double angle formula to express first term in terms of just q instead of 2q
3. Get one side equal to zero and factor out any common trig functions or reverse FOIL.
4. See if equation is in quadratic form and reverse FOIL. (Replace the trig function with x to see how it factors if that helps.)
5. If the angle you are solving for is a multiple of q, don't forget to add 2 to your answer for each multiple of q since q will still be less than 2 when solved for.
HELPFUL HINTS FOR SOLVING TRIGONOMETRIC EQUATIONS
2. Be on the look-out for ways to substitute using identities
7) 8)
NO solution forcos q = 3.
ReferenceAngle
Solving Trigonometric Equations
sin q 12 q
6
q 0 7
611
6, , ,
cosq 12
or
q
353
,
2 02sin sinq q+ sin ( sin )q q2 1 0+ sin sinq q + 0 2 1 0or
( cos )(cos )2 1 3 0q q- - 2 7 3 02cos cosq q- +
2 1 0 3 0cos cosq q- - orcosq 3
9) Solve 6 sin2 x + sin x – 1 = 0; 0º ≤ x < 360º
A quadratic equation!
It may help to abbreviate sin x with s:
i.e. 6s2 + s – 1 = 0
Factoring this: (3s – 1)(2s + 1)= 0
1 1sin or sin3 2
x x -∴
α = 19.47º α = 30º
So, x = (To nearest 0.1º.)19.5°, 160.5°, 210°, 330°.
You Try…Solve for y
03sin13sin12 2 +- yy
01sin33sin4 -- yyy
y
yy
-
43arcsin
43sin
3sin403sin4
y
y
yy
-
31arcsin
31sin
1sin301sin3
3398.0y 8481.0y
10)
01334
031312 2
--
+-
uu
uu
Solve for y: domain
We have to give all the answers
3398.0
8481.0
)2,0[
31
43 ?
?
294.28481.0 -
803.2339.0 -
11)The equation cannotbe factored. Therefore,use the quadratic equation to find the roots:
Reference Angles:
Solving Trigonometric Equations
tanq - -b b ac
a
2 42
tan ( ) ( ) ( )( )( )
q - - - - -1 1 4 3 1
2 3
2
3 1 02tan tanq q- -
tan . tan .q q- 0 43 0 76or
q 0 4061. q 0 6499.
q 2 7355 5 877 0 6499 3 7915. , . , . , .
3 tan2q - tanq - 1 0
Using a Graphing Calculator to Solve Trigonometric Equations
Therefore, q 0.654, 2.731, 3.796, and 5.873 .
2 2cos sin sin 0q q q- + Use the Pythagorean Identity to replace this with an equivalent expression using sine. 2 2cos 1 sinq q -
2 21 sin sin sin 0q q q- - + Combine like terms, multiply by -1 and put in descending order22sin sin 1 0q q- - Factor (think of sin q like x
and this is quadratic)
(2sin 1)(sin 1) 0q q+ - 1sin , sin 12
q q-
Set each factor = 0 and solve7 11, ,
6 6 2 q
12)
Solve for x where the domain is )2,0[ 2tansincos + xxx
35,
3;
21arccos
cos21;2
cos1
xx
xx
13)
2cos
sincos
2cossin
coscos
2cossinsincos
22
22
+
+
+
xxx
xx
xx
xxxx
Use a graphing utility to solve the equation. Express any solutions rounded to two decimal places.
3sin1722 - xxGraph this side as y1
in your calculatorGraph this side as y2
in your calculator
You want to know where they are equal. That would be where their graphs intersect. You can use the trace feature or the intersect feature to find this (or these) points (there could be more than one point of intersection).
There are some equations that can't be solved by hand and we must use a some kind of technology.
14)
3sin1722 - xx This was graphed on the computer with graphcalc, a free graphing utility you can download at www.graphcalc.com
After seeing the initial graph, lets change the window to get a better view of the intersection point and then we'll do a trace.
Rounded to 2 decimal places, the point of intersection is x = 0.53
check: 22 .53 17sin .53 3.066 3-
This is off a little due to the fact we approximated. If you carried it to more decimal places you'd have more accuracy.
YOU TRY…Solve
cos cos
cos cos: ( cos )(cos )
( cos ) (cos )
( cos ,cos ) (cos )
( , ) ( )
x x
x xfactor x x
x OR x
x x OR x
x n n OR x n
+ -
+ + + +
+ + -
- -
+ + +
2
22 3 12 3 1 0
2 1 1 02 1 0 1 0
12 1 122 42 2 23 3
xx 2sin23cos3 .1 +-0cos33)cos1(2 2 --- xx
2cot cos 2cotx x x2cot cos 2cot 0x x x-
2cot cos 2 0x x - 2cot 0 cos 2 0x or x -
3,2 2
x
2cos 2x 2cos 2x
cos 2x
x
Note: There is no solution here because 2 lies outside the range for cosine.
2)
Closure
Discuss the similarities and differences in the steps for solving a trigonometric equation versus solving a polynomial equation.
Solving Trigonometric Equations with Multiple Angles
1cos32
x
53 ,3 3
x
5,9 9
x
Solve:
Solution:
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.
Return to the step where you have 3x equalingthe two angles and find coterminal angles for those two.
7 11, ,3 3 53 ,
3 3x
Divide those two new angles by 3.7 11, ,9 9 5,
9 9x
1)
Solving Trigonometric Equations
The solutions still do not exceed 2. Return to 3x and find two more coterminal angles.
13 17, ,3 3 7 11, ,
3 3 53 ,
3 3x
Divide those two new angles by 3.13 17, ,9 9 7 11, ,
9 9 5,
9 9x
The solutions still do not exceed 2. Return to 3x and find two more coterminal angles.
19 23, ,3 3 13 17, ,
3 3 7 11, ,
3 3 53 ,
3 3x
Divide those two new angles by 3.19,913 17, ,
9 9 7 11, ,
9 9 5,
9 9x
Notice that 19/9 now exceeds 2 andis not part of the solution.
Therefore the solution to cos 3x = ½ is 5 7 11 13 17, , , , ,
9 9 9 9 9 9x
2) Solve the equations.
Example 3) Solve 2sin 3 42x+
0 .2
x2sin 1
21sin
2 2
x
x
5 and 6 6
5 or 2 6 2 6
5 or 3 3
x x
x x
x
y
Take the fourth root of both sides to obtain: cos(2x)= ±
23
From the unit circle, the solutions for 2q are 2q = ± + kπ, k any integer.
π 6
Example 4: Find all solutions of cos4(2x) = .9 16
Answer: q = ± + k ( ), for k any integer. 12
π 2
π
1 π 6-π 6
x = -23 x =
23
ππ
5) Solve 32
3 0tan x+
Example 6: Solve 3 + 5 tan 2x = 0; 0º ≤ x ≤ 360º.
Firstly we need to make tan 2x the subject of the equation:
The tangent of an angle is negative in the
- 5
3tan α 1 = 30.96°
The range must be adjusted for the angle 2x. i.e. 0° ≤ 2x ≤ 720°.
Hence: 2x =
x = 74.5°, 164.5°, 254.5°, 344.5°.
149.04°, 329.04°, 509.04°, 689.04°.
2nd and 4th quadrants.
tan 2x = – 35
Example 7: Solve 2 sin (4x + 90º) – 1 = 0; 0 < x < 90º 12
The sine of an angle is positive in the
The range must be adjusted for the angle 4x + 90º.i.e. 0º < 4x < 360º 90º < 4x + 90º < 450º
4x + 90º =
4x = 60º, 300º
x = 15º, 75º
-
21sin α 1 = 30º
150°, 390°
1st and 2nd quadrants.
sin (4x + 90º)
Solving Trigonometric Equations
24sin 2cos 1x x +
csc cot 1x x+
3sin 22
x -
Try these:
2cos2 2x
5.4218x
2x
2 5 5 11, , ,3 6 3 6
x
2x
1.
2.
3.
4.
Solution
Sect. 5-3#’s 61, 63, 65,
Evaluate.
a) tan2 116 b) sec2 2
3 -
33
2
13
- 2 2
4
Finding Exact Values
Solving Trigonometric Equationscos 1 sinx x+ 2 2cos 1 sinx x+
2 2cos 2cos 1 sinx x x+ +
Solve:
2 2cos 2cos 1 1 cosx x x+ + -22cos 2cos 0x x+
2cos cos 1 0x x + 2cos 0 cos 1 0x or x + cos 0x cos 1x -
3,2 2
x x X
Why is 3/2 removed as a solution?
26)
5.5 Trigonometric Equations• Objectives
–Find all solutions of a trig equation–Solve equations with multiple angles–Solve trig equations quadratic in form–Use factoring to separate different functions in
trig equations–Use identities to solve trig equations–Use a calculator to solve trig equations
Is this different that solving algebraic equations?• Not really, but sometimes we utilize trig
identities to facilitate solving the equation.• Steps are similar: Get function in terms of one
trig function, isolate that function, then determine what values of x would have that specific value of the trig function.
• You may also have to factor, simplify, etc, just as if it were an algebraic equation.
2cos 1 0q - 5)2cos 1x
1cos 2x
5,3 3
x
QI QIV
8)
ReferenceAngle
sin q 12
or sin q - 1
q
6
q
656
32
, ,
2 1 0 1 0sin sinq q- + or
2 1 02sin sinq q+ - ( sin )(sin )2 1 1 0q q- +
14)
Reference Angle:
Therefore:
sin q 13
4 1sin sinq q- 3 1sinq
q 0 3398.
q 0 3398 2 8018. .and