Solve the equation 2cos2x° - 1 = 0 0 ≤x ≤ 360
I really dont undersand this kind of questions, so can someone go step by step, to show me how to do it.
I really dont undersand this kind of questions, so can someone go step by step, to show me how to do it.
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2cos2x° - 1 = 0 becomes 2cos2x = 1 and so cos2x = 1/2.
Now if 0 ≤x ≤ 360 then 0 ≤2x ≤ 720, so we need to consider all solutions up to 720 to start with because of the 2x.
Cos is positive in the 1st and 4th quadrants.
So 2x = 60 and 360 - 60 giving 2x = 60 and 300.
Now add 360 on to each of these to get the 4 solutions for 2x
We have 2x = 60, 300, 420 and 660.
Divide by 2 to get x = 30, 150, 210 and 330. Hope you can follow this!
Now if 0 ≤x ≤ 360 then 0 ≤2x ≤ 720, so we need to consider all solutions up to 720 to start with because of the 2x.
Cos is positive in the 1st and 4th quadrants.
So 2x = 60 and 360 - 60 giving 2x = 60 and 300.
Now add 360 on to each of these to get the 4 solutions for 2x
We have 2x = 60, 300, 420 and 660.
Divide by 2 to get x = 30, 150, 210 and 330. Hope you can follow this!
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1) Let's start like you would a normal equation, without any trigonometric functions. Here, lets rename the term "cos2x" as "A" just to make it look more familiar! We now have:
2A - 1 = 0
2A = 1
A = 1/2
Easy enough? Just like algebra!
2) So now we need to but that cos2x back in! Using what we solved for in (1)
cos2x = 1/2
3) Now we can use inverse cosine (seen in your notes probably as cos to the -1 power) to isolate x on one side!
invcos( cos 2x) = invcos (1/2)
2x = invcos (1/2)
x = 1/2 invcos (1/2)
4) Now you need to understand what inverse cosine is. Inverse cosign is asking "What angle gives a cosine of 1/2?" Or, using your unit circle-- at what angle is the x value 1/2?
5) Now we have to consider any non obvious answers. We know x is between 0 and 360 degrees, so 2x mus be between 0 and 720 degrees. Are there any other answers that ALSO give us 1/2. Or-- what point on the unit circle has the same x value?
That is at 300 degrees, 420 degrees, and 660 degrees!
6) Now remember, we need to go back to that final equation in step 2. We know that invcos(1/2) is 60 and 300. But
x=1/2 invcos(1/2)
So, x is equal to 30, 150, 210, and 330! Final answer :)
You should always double check that by plugging x back in!
If you had trouble with step 1, you should go back and look at algebra, and solve some simple equations to get the hang of it. If you had problems with steps 3, I would take another look at inverse cosine. Trouble with step 4 and 5, I would review the material on sine, cosine, and the unit circle!
Good luck!
2A - 1 = 0
2A = 1
A = 1/2
Easy enough? Just like algebra!
2) So now we need to but that cos2x back in! Using what we solved for in (1)
cos2x = 1/2
3) Now we can use inverse cosine (seen in your notes probably as cos to the -1 power) to isolate x on one side!
invcos( cos 2x) = invcos (1/2)
2x = invcos (1/2)
x = 1/2 invcos (1/2)
4) Now you need to understand what inverse cosine is. Inverse cosign is asking "What angle gives a cosine of 1/2?" Or, using your unit circle-- at what angle is the x value 1/2?
5) Now we have to consider any non obvious answers. We know x is between 0 and 360 degrees, so 2x mus be between 0 and 720 degrees. Are there any other answers that ALSO give us 1/2. Or-- what point on the unit circle has the same x value?
That is at 300 degrees, 420 degrees, and 660 degrees!
6) Now remember, we need to go back to that final equation in step 2. We know that invcos(1/2) is 60 and 300. But
x=1/2 invcos(1/2)
So, x is equal to 30, 150, 210, and 330! Final answer :)
You should always double check that by plugging x back in!
If you had trouble with step 1, you should go back and look at algebra, and solve some simple equations to get the hang of it. If you had problems with steps 3, I would take another look at inverse cosine. Trouble with step 4 and 5, I would review the material on sine, cosine, and the unit circle!
Good luck!
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cos 2x = 1/2
2x = 60° , 300° , 420° , 630°
x = 30° , 150° , 210° , 315°
2x = 60° , 300° , 420° , 630°
x = 30° , 150° , 210° , 315°