Solve csc(4x) − 4 = 0 for the four smallest positive solutions Thank you!
1 answer:
Answer:
x ∈ {0.063170, 0.722228, 1.633966, 2.293024}
Step-by-step explanation:
It is probably convenient to replace csc(4x) with its equivalent, 1/sin(4x). Then add 4 and multiply by sin(4x)/4.
1/sin(4x) -4 = 0
1/sin(4x) = 4
1/4 = sin(4x)
Using the inverse sine function, we find ...
4x = arcsin(1/4) +2kπ . . . . in radians
and the supplementary solution, ...
4x = π -arcsin(1/4) +2kπ
Dividing these by 4 gives the possible values of x.
x = arcsin(1/4)/4 +kπ/2
x = π/4 -arcsin(1/4) +kπ/2
The four smallest positive solutions will be found where k = 0 or 1.
x ∈ {0.063170, 0.722228, 1.633966, 2.293024}
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I like a graphing calculator for solving these quickly and easily.
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<h3>Answer: 93 degrees</h3>
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Explanation:
Check out figure 1 which is one of the attached images below.
In this diagram, I have angle A as 75 degrees and angle B as 51 degrees.
Angle C is therefore, C = 180-A-B = 180-75-51 = 54 degrees.
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Point D is somewhere between A and B such that it is on segment AB.
Figure 2 (also attached as an image) shows segment CD forming two angles DCB and ACD.
These are the blue and red angles respectively, such that the blue angle is twice as large as the red angle.
blue angle = 2*(red angle)
This is what it means when it says the ratio of the two angles is 2:1.
I have 2x as the blue angle and x as the red angle. We don't know what x is yet, but we do know that the x and 2x combine back to angle C = 54 degrees.
So,
(angle DCB) + (angle ACD) = angle C
(2x) + (x) = 54
3x = 54
x = 54/3
x = 18
Since x = 18, this means 2*x = 2*18 = 36
Therefore,
angle DCB = 2x = 36 degrees
angle ACD = x = 18 degrees
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Focus solely on triangle DCB. We found angle DCB = 36 degrees and we know that angle DBC = 51
The remaining angle y = angle BDC is...
(angle BDC)+(angle DCB)+(angle DBC) = 180
(y)+(36)+(51) = 180
y+87 = 180
y+87-87 = 180-87
y = 93
angle BDC = 93 degrees
Figure 3 shows the angles we found (basically I replaced x, 2x and y with their respective numbers).