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5.8
Trigonometric Equations
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Solving linear and quadratic trigonometric
equations on the interval 0≤x≤2π |
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Team members |
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Name: Anum Khan
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Overview |
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This section
shows, with detailed explanations, how to solve
trigonometric equations using different methods
and strategies and the properties of
trigonometric functions and identities. A
trigonometric equation is any equation that
involves one or more trigonometric ratios of a
variable.
Examples of
Trigonometric Equations include:
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3 sin
θ+ cos
θ=0 NOT
2x + 5 tan 4= 1.5
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θ +sin θ=2
By the end of this
section, one should be able to solve linear and
quadratic trigonometric equations on the
interval 0≤x≤2π. Some operations may even
include irrational terms such as square roots.
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Study Tips,
Methods and or Advice |
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You may use some
useful study tips that may help you solve your
trigonometric equations faster and easier.
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Using
the CAST rule helps solving trigonometric
equations as it helps identify what quadrant
will make your function positive on the grid.
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Also,
drawing a quadrant to facilitate visually and
check over would be a good idea.
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Using
a table of common sin, cos, and tan values will
help solve equations limiting the time spend for
each operation.
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It is
very useful to use the following trigonometric
properties to determine other angles that fall
within the required range:
1. sin
θ =sin(180°- θ)
2. cos
θ=cos(360°- θ)
3. tan
θ= tan(180°+ θ)
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Sample
Questions and complete solutions |
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Solve the following
equations for the interval 0≤x≤2π.
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Questions |
Complete Solutions |
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1.
sin
θ = 1
2
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2
Since
sin is positive, it is located in the 1st
or 2nd quadrant.
θ
θ
In order
to obtain a positive value, both θ have
to be in the range of 0 to
.
Thus
the exact solutions are
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2. Cos
θ=
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Since
cos θ is negative, θ is in quadrant 2 or
3.
Thus
θ should be
in multiples of 4.
Thus
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3.Factor:
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Let “m”
represent “Cos
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Factor by
Grouping
m( 3m+1) – (3m+1)=0
(3m+1)(m-1)=0
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4. 6sin
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Rearrange the
Pythagorean Identity:
Multiply the
equation by (-1)
Factor by
Grouping
Either
(θ
is negative :quadrant I or II)
OR
Thus , the
exact solutions are
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Extra
Practice Questions and Answers |
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Solve the following for
0≤x≤2π.
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Extra Practice Questions |
Answers |
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1. a) sin
θ= 0
b) cos θ= 0 c) tan θ= 1
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a) 0, π,
2π b) π , 3π
2 2
c)
π , 5π
4 4 |
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2.a) sin
θ= -1
b)
c)
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a)
b)
,
c)
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3.
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4.
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(2sin-3)(sin+2) |
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5.
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0.73,2.41,
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Self
Reflection |
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On the
whole, even though this section of trigonometry
may seem as the most difficult, there is an
advantage to trigonometric equations since it
connects ideas from other chapters in the
functions and relations curriculum. This
includes areas such as graphing, solving for
various values of a variable, determining
restrictions, and so forth. Personally, I had a
hard time while learning trigonometric
equations, but when you relate it to other
chapters in the book such as chapter 2 which
deals with rational expressions, it becomes much
easier to understand and solve problems. The key
to doing well in this chapter is to keep some
necessary equipment, such as a chart including
common sin, cos and tan values as well as
repeatedly required formulas, in hand. Making
sure to keep notes organized also helps to
improve study patterns and a better grade.
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