Expert: Sherry Wallin - 11/26/2010

Question
QUESTION: Can you please help me.  Please!

Can you please help me.

Solve 4sin(square)A + 1 = sin(square)A + 2 to the nearest ten minutes or nearest tenth of a degree in the interval  ( 0º less than or equal to A less than 360º)

Thank you very much!


ANSWER: Kamla~

There are a number of ways to do this kind of problem. One is to use substitution so that it looks like a problem you feel comfortable doing, so let sin A = x so that sin^2A = x^2 and your problem now looks like:

4x^2 + 1 = x^2 + 2
3x^2 + 1 = 2          subtract x^2 from each side
3x^2 = 1          subtract 1 from each side
x^2 = 1/3          divide by 3 each side
x = +-sqrt(1/3)        take square root on each side
x = +-1/sqrt(3)        square root of 1 is 1
x = +-sqrt(3)/3        rationalized the denominator
Sin A = +-sqrt(3)/3    substitute back in for Sin A = x

You can take the inverse (arcsin) on each side to find A or you can ask yourself what angle A returns the sin A = +-sqrt(3)/3?
Since the values +-sqrt(3)/3 are not one of the 'typical' angles you will need to use the arcsin(sqrt(3)/3) and find it to be 35.3 deg and then use your knowledge of where the sin A is negative (quadrants 3,4) and then you know that this will occur at 180 + 35.3 deg = 215.3 deg and at 360 - 35.3 = 324.7 deg

Math Prof


---------- FOLLOW-UP ----------

QUESTION: As θ increases from π to 3/2π, which statement is true?
a)cos θ increases from -1 to 0
b)sin θ increases from -1 to 0
c)sin θ decreases from 1 to 0
d)cos θ decreases from 1 to 0


Answer
The sin θ at 0 deg is 0 and then at 90 deg is 1 and at 180 deg is 0 again and at 270 deg is -1 and at 360 deg = 0 deg is 0 again. The
cos θ acts the opposite of the sin θ. At 0 deg the cos θ is 1 and at 90 deg is 0 and at 180 deg is -1 and at 270 deg is 0 again and back to 360 deg is 1 again. The sin θ increases as the cos θ decreases and vice versa depending on which quadrant you are in and at 45 deg their values are the same (or any multiple of 90 deg added to 45 deg such as 45 + 90 = 135 deg or 45 deg + 180 deg = 225 deg. I purposely did not give you an exact answer to your question because you need to think about this relationship between the sin θ and the cos θ and if I just tell you the answer that won't help you to learn the relationship between the sin θ and the cos θ. The answer is, of course, here but you will have to figure out where.


Math Prof