Calculate angle from sine and cosine

[batterypack]

sin x = 0.5299
cos x = 0.8480

Without using inverse cos or inverse sin, is it possible to calculate the angle?

 

[HallsofIvy]

No, that's what the inverse functions are for! (There may be other ways to approximate the value of the angle but if you want the correct value, you would have to use either inverse sine or inverse cosine.)

 

[batterypack]

Misunderstood this question:

"Suppose you are told that the sine of a certain angle is 0.5299 and the cosine of the same angle is 0.8480. What is the sine of twice this angle. Don't use the trigonomtric functions keys on your calculator to figure this out."

I have the answer, but no idea how to do this.

 

[batterypack]

Found it, sin 2a = 2 (sin a) (cos a)

Can this be generalised further? What if you want to find 3 or 4 times the sine of the angle?

 

[Mark44]

I'll answer the question you asked (which is probably not the question you meant). If you are given the sine of an angle, then 3 times the sine of the angle is 3*sin(a), and 4 times the sine of the angle is 4*sin(a).

My point in saying this was to get you to think about what you're asking.

Assuming you really meant the sine of 3 or 4 times the angle, then yes, there are identities that can be used.

sin(3a) = sin(2a + a) = sin(2a)cos(a) + cos(2a)sin(a) = 2sin(a)cos(a)*cos(a) + (cos^2(a) - sin^2(a))sin(a).

You can break down sin(4a) to sin(2a + 2a) and continue working with that.

 

[batterypack]

Yes, I meant the sine of 3 or 4 times the angle. Thanks.