Trigonometry
Sec. 02 notes
MathHands.com
Márquez
Solving Trig Equations: The Almost-Easy Ones
Main Idea
In the last section we solved equations such as:
Solve
tan (x) =
6
5
Solution:
First we graph y = 1.2 then we graph y = tan(x), then we mark the intersections. These are the solutions. Clear
from the graph is that we have infinite many of them.
Of these, the first one is determined by using a calculator to estimate tan−1 (1.2) ≈ 50.194◦. This solution is the
only one provided by the tan−1 function.
y=
5
6
5
y = tan(x)
4
3
2
1
-360◦ -315◦ -270◦ -225◦ -180◦ -135◦ -90◦
-45◦
.194
45◦
90◦
135◦ 180◦ 225◦ 270◦ 315◦ 360◦
-1
-2
-3
-4
≈ −309.806◦
180◦
≈ −129.07◦
180◦
-5
≈ 50.5◦
180◦
≈ 231◦
180◦
We conclude the solution by describing all possible values of x.
x ≈ 50.194◦ + k180◦
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for k ∈ Z
pg. 1
Trigonometry
Sec. 02 notes
MathHands.com
Márquez
Now consider if in the equation
6
5
the x was replaced with something else, such as θ. One could solve it the same manner, with the same results:
tan(x) =
Example:
Solve
tan (θ) =
6
5
Solution:
The solution for possible values of θ:
θ ≈ 50.194◦ + k180◦
for k ∈ Z
Example:
Solve
tan (blah) =
6
5
Solution:
The solution for possible values of blah:
blah ≈ 50.194◦ + k180◦
Example:
Solve
tan
=
for k ∈ Z
6
5
Solution:
The solution for possible values of
:
≈ 50.194◦ + k180◦
for k ∈ Z
Now the punch-line...
Example:
Solve
6
tan 2x + 30◦ =
5
Solution:
From solving the easy version of the equation we obtain
2x + 30◦ ≈ 50.194◦ + k180◦
for k ∈ Z
Therefore:
2x + 30◦ ≈50.194◦ + k180◦
2x ≈20.194◦ + k180◦
x ≈20.194◦ + k90
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2007-2009
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pg. 2
Trigonometry
Sec. 02 notes
Example:
Solve
MathHands.com
Márquez
4
tan 3x + 45◦ = −
5
Solution:
From solving the easy version of the equation we obtain
3x + 45◦ ≈ −38.66◦ + k180◦
for k ∈ Z
Therefore:
3x + 45◦ ≈ − 38.66◦ + k180◦
3x ≈ − 83.66◦ + k180◦
x ≈ − 83.66◦ + k60
Example:
Solve
7
tan 2x − 45◦ =
3
Solution:
From solving the easy version of the equation we obtain
2x − 45◦ ≈ 66.801◦ + k180◦
for k ∈ Z
Therefore:
2x − 45◦ ≈66.801◦ + k180◦
2x ≈111.801◦ + k180◦
x ≈111.801◦ + k90
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2007-2009
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pg. 3
Trigonometry
Sec. 02 exercises
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Márquez
Solving Trig Equations: The Almost-Easy Ones
1. Solve
7
tan 2x − 45◦ =
3
2. Solve the equation
sin x =
1
2
3. Solve the equation
sin(2t − 50◦ ) =
4. Find all solutions
sin(2θ) =
1
2
−1
2
5. Find all solutions
sin(3θ + 90◦ ) = 0
6. Find all solutions
sin(2x − 40◦ ) =
7. Find all solutions
cos(5x + π) =
−1
2
cos(2t − π) =
−1
3
8. Find all solutions
9. Find all solutions
csc
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2007-2009
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1
3
2x + π
3
math
hands
= −2
pg. 4