After solving a trig equation, how do you find all answers in the interval [0, 2pi)?

Question

I know it has something to do with reference angles but idk. Right now I have -30 degrees

Answer 1

So you found the first angle...you are half of the way done. Now to find the other angle.

First, you need to think about what trig function you are working with. Let's suppose for this example you had the equation sin x = 1/2. Taking the inverse sin of both sides would tell you that x = 30 degrees, {on a calculator you would just type sin^-1(1/2) }. But is that the only place where the sin of x would equal 30 degrees? No...and that's where your idea of a reference angle comes into play.

Take a minute to recall which trig functions are positive in which quadrants. All trig functions are positive in Q1, sin is positive in Q2, tan in Q3, and cos in Q4. This info will help you find your reference angle.

To find the reference angle, you basically take your found angle, see what quadrant it is in, and see if your given trig function is positive or negative in that quadrant that the angle was found in. Whatever other quadrant the same trig function is either positive of negative in is the quadrant our reference angle is in. All we do to get the reference angle is measure our found degrees off of the x-axis into the quadrant that coincides with the sign of our given trig function. I know that sounds like a lot, but it's not too bad...take a look at our example...

In our example, we had sin x = 1/2 and found x to equal 30 degrees. Well, 30 degrees is in Q1. Now look at the sign (+/-) of sin in Q1. Sin is positive in Q1. So we ask ourselves, where else is sin positive...oh, it's also positive in Q2. So look at the x-axis that borders Q2...that is at 180 degrees. Measuring up into Q2 off of the x-axis by our found angle of 30 degrees gives us 180 - 30 = 150 degrees (we subtract because if we are going into Q2 from x-axis, we are decreasing the angle from 180). Now we can convert our degree measure to radians (as you noted we need to be in the interval [0, 2pi). To do this, you simply multiply your degree by (pi/180) and simplify. So 30(pi/180) = (30/1)(pi/180) = 30pi/180 = pi/6. Likewise, 150(pi/180) = 150pi/180 = 5pi/6. So your answers would be pi/6 and 5pi/6. (Realize that pi and 180 degrees are essentially the same thing, so to multiply by pi/180 is like mulitplying by 1...we don't change the original number when we multiply by 1, we just change the way it looks)

Lets look at another example.

Suppose your original equation was cos x = 1/2. Taking the inverse cosine of both sides {on calculator you type cos^-1(1/2) }, yields x = 60 degrees. OK, so again our found angle is in Q1, and cosine is positive in Q1. Where else is cosine positive...oh in Q4...so that's where our reference angle would be found. So we measure off of the x-axis into Q4. Well, the x-axis bordering Q4 can be looked at as 0 degrees or 360 degrees...since we will be going back into Q4, making the angle smaller, we will use 360 degrees, so to avoid using negative angles. So, 360 - 60 = 300 degrees, and this is our second answer. Again, convert into radians. 60(pi/180)= 60pi/180 = pi/3. 300(pi/180) = 300pi/180 = 5pi/3. And that's it.

Had our found angle been negative, we would have looked to see where our trig function was also negative and then proceded in the same fashion.

There will almost always be two solutions to these problems. The only time there will be one solution is if sin x or cos x = 1 or -1. Then the angles lie on the bondries of the quadrants and are called "quadrangles". The only times you will have no solutions is if either sin x or cos x if greater than 1 or less than -1. This cannot be, as all values in the range of sin and cos are between -1 and 1...but that's another lesson. For now, just try to understand how to use your found angle to get yoru reference angle.

I hope this has helped. Good luck.

Answer 2

There is the smart way to do this and the lazy way. The lazy way is to keep adding and subtracting multiples of the answer you have and testing each one out until you get to 360 degrees or negative 360 depending on if you had been adding or subtracting.

The smart way to go about it is to try to use some knowledge of the problem to calculate what the angles will be, but if you knew that you wouldn't be asking this question i the first place. just keep plugging in -30,0,30,60,90,120,150,180 etc until things work.

Answer 3

use the formula or anagram All Silver Tea Cups
All positive in the first quadrant that is 0-90*
Sine and Cosecant positive in the second quadrant that is 90-180*
tangent and cotangent positive in the third quadrant that is 180*-270*
cos and sec positive in the fourth quadrant that is 270-360*
if you get the solution as x the other solution will be as per the following rule
for sin the two values will be x and 180-x
for cos it will be x and 360-x
for tan and cot it will be x and 180+x and

Answer 4

And, from the way the question was written, in addition to the -30 degrees you also have 'not a clue'.

How about letting us in on what the problem was to begin with?


Doug

Answer 5

Well -30 degrees is the same as ( -1pi/6 ) then it would just depend on what are you looking for sine cosine or tangent. it would either be (-5pi/6), ( -7pi/6 ), or (-11pi/6).

Answer 6

ask your mom

Answer 7

do your own homework.