trogonometry?

Question

1. derive the identity for sin 3x in terms of sin x

2. using the double-angle formula, find sin 120*

Answer 1

1) sin(3x)

To solve this, you need to know two things:
a) The sine addition identity, sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
b) The double angle identities, sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x)

With that in mind,

sin(3x) = sin(2x + x)
= sin(2x)cos(x) + sin(x)cos(2x)
= 2sin(x)cos(x)cos(x) + sin(x) ( cos^2(x) - sin^2(x) )
= 2sin(x)cos^2(x) + sin(x) cos^2(x) - sin^3(x)

Now, use the identity cos^2(x) = 1 - sin^2(x), to obtain

= 2sin(x) (1 - sin^2(x)) + sin(x) ( 1 - sin^2(x) ) - sin^3(x)
= 2sin(x) - 2sin^3(x) + sin(x) - sin^3(x) - sin^3(x)
= 3sin(x) - 4sin^3(x)

2) sin(120)

The double angle formula isn't really required, because 120 degrees = 2&pi/3, and 2&pi/3 is a known unit circle value. However, we can still use that formula in this way:

sin(120) = sin(2*60)
= 2sin(60)cos(60)
= 2 ( sqrt(3)/2 ) ( 1/2)
= sqrt(3)/2

Are you sure you didn't mean something else, like 12 degrees?

Answer 2

1.
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
so, sin(3x) = sin(2x + x) = sin(2x)cos(x) + cos(2x)sin(x)

2.
sin(2x) = 2sin(x)cos(x)
so, sin(120*) = sin(60*+60*) = 2sin(60*)cos(60*)

Answer 3

Is "trogonometry" higher math for cavemen?