8.5 Solving More Difficult Trigonometric Equations

1
8.5 Solving More Difficult Trigonometric
Equations
Objective To use trigonometric identities or
technology to solve more difficult trigonometric
equations.
2
Example 1 To find all solutions of cos4(2? )
.
3
Some trigonometric equations can be transformed
into equations that have quadratic form.
Example 2 Find all solutions of the
trigonometric equation tan2 ? tan ?
0.
Therefore, tan? 0 or tan? -1.
The solutions for tan ? 0 are the values ?
kp, for k is any integer.
4
Example 3 The trigonometric equation 2
cos2? 3 sin? 3 0 .
Solution Use the Pythagorean identity 2 (1
sin2? ) 3 sin? 3 0 .
2 sin2? 3 sin? 1 0 implies that
(2 sin? 1)(sin? 1) 0.
Therefore, 2 sin? 1 0 or sin? 1 0.
? -p/2 2kp, from sin ? -1
5
Steps for Solving

• Isolate the Trigonometric function.
• Then solve for the angle exactly if the ratio
  represents known special triangle ratios.
• Or solve for the angle approximately using the
  appropriate inverse trigonometric function.

6
Complete the List of Solutions
If you are not restricted to a specific interval
and are asked to give a complete list of
solutions (general solution), then remember that
adding on any integer multiple of 2p represents a
co-terminal angle with the equivalent
trigonometric ratio.
7
Example 4 Solve 8 sin ? 3 cos2 ? with ? in
the interval 0, 2p.
Solution Rewrite the equation in terms of only
one trigonometric function.
Therefore, 3 sin? ? 1 0 or sin? 3 0
8
(No Transcript)
9
Example 5 5cos2? cos ? 3 0 for 0 ?
p.
Solution The equation is quadratic. Let u
cos? and solve 5u2 u ? 3 0.
Therefore, cos? 0.6810249 or 0.8810249.
Use the calculator to find values of ? in 0 ?
p.
This is the range of the inverse cosine function.
The solutions are ? cos ?1(0.6810249 )
0.8216349 and ? cos ?1(?0.8810249) 2.6488206
10
Example 6 Solve a trigonometric equation by
factoring.
Solution
11
Example 7 Solve tan2x 3tanx 4 0 in 0
x 2?. Solution Let u tan x, then x2
3x 4 0 (x 4)(x 1) 0 x 4 0
or x 1 0 x 4 or x 1 tan
x 4 or tan x 1 x arctan4 or
x 3?/4 (x 76 or x 135) x
? arctan4 or x 7?/4 ( x 256
or x 315)
12

• Example 8 Solve. 3sinx 4 1/sinx in 0
  x 2? .
• Solution Let x sin x, then
• 3x 4 1/x
• x(3x 4) x(1/x)
• 3x2 4x 1
• (3x 1)(x 1) 0
• 3x 1 0 or x 1 0
• x 1/3 or x 1
• sin x 1/3 or sin x 1
• x arcsin(1/3) or x 3?/2
• ( x 19 or x 270)
• x ? arcsin(1/3)
• ( x 161)

13
Solution
14
Solution
Particular solutions are
General solutions are
15

• Example 11 Solve 2 cos x sec x 0
• Solution

Therefore,
or
However,
So,
is not a real number, thus the equation has no
solution.
16

• Example 12 Solve cos x 1 sin x in
  0, 2?
• Solution

Like solving algebraic equation, once we squared
the original trigonometric equation, it may
generate some extraneous solution. We need to
check the solutions.
Since we squared the original equation we have to
check our answer. The only solutions are ?/2 and
?.
17
Since we squared the original equation we have to
check our answer.
18
Since we squared the original equation we have to
check our answer.
19
Using a Graphing Calculator to Solve
Trigonometric Equations
0 ? ? ? 2?
a)
b)
The equation cannot be factored. Therefore, use
the quadratic equation to find the roots
Reference Angle
Therefore
Reference Angles
20
Using a Graphing Calculator to Solve
Trigonometric Equations
Therefore, q 0.654, 2.731, 3.796, and 5.873 .
5.2.13
21
Remark Solving trigonometric equations is a long
lasting task through out the entire trigonometry.
After we learned the sum and difference of
angles, double angles, triple angles, we will be
able solve some much more difficult trigonometric
equations.
22
Assignment P. 326 1 22