# Trignometric Functions

• May 13th 2007, 03:43 PM
cuteisa89
Trignometric Functions
Can somebody help me with these?

1. Simplify the expression sinx (cscx - sin x)
2. rewrite the expression cos^2 / (1 + sin x) so that there is no fraction
3. use the half-angle formula to simplify the expression sqrt ((1+cos(4x))/2)
4. Use the following to find the exact value of sin(x/2). cosx = -4/5, x is in quadrant III
5. Evaluate the expression sin^2(355pi/ 113) + cos^2 (355pi/113)
6. verify the identity cos x/(1 + sinx) + cosx/(1-sinx) = 2 secx

Thank you!
Isabel
• May 13th 2007, 06:31 PM
Jhevon
Quote:

Originally Posted by cuteisa89
1. Simplify the expression sinx (cscx - sin x)

sinx(cscx - sinx) = sinx(1/sinx - sinx)
.......................= sinx/sinx - sinx*sinx
.......................= 1 - sin^2x
.......................= cos^2x

Quote:

2. rewrite the expression cos^2 / (1 + sin x) so that there is no fraction
cos^2x/(1 + sinx) = (1 - sin^2x)/(1 + sinx) ...the top is the difference of two squares
.........................= [(1 + sinx)(1 - sinx)]/(1 + sinx)
.........................= 1 - sinx

Quote:

3. use the half-angle formula to simplify the expression sqrt ((1+cos(4x))/2)
by the half angle formula:

cos(x/2) = sqrt[(1 + cosx)/2]

if we replace x with 4x we get:

cos(2x) = sqrt[(1 + cos(4x))/2]

so cos(2x) is the simplified expression

Quote:

4. Use the following to find the exact value of sin(x/2). cosx = -4/5, x is in quadrant III
sin(x/2) = - sqrt[(1 - cosx)/2] ..........sine is negative in quad III
we are given that cosx = -4/5
=> sin(x/2) = -sqrt[(1 - (-4/5))/2]
................= -sqrt[(9/5)/2]
................= -sqrt(9/10)
................= -3/sqrt(10)
................= -3sqrt(10)/10

Quote:

5. Evaluate the expression sin^2(355pi/ 113) + cos^2 (355pi/113)
sin^2(355pi/ 113) + cos^2 (355pi/113) = 1
since sin^2x + cos^2x = 1 for all x

Quote:

6. verify the identity cos x/(1 + sinx) + cosx/(1-sinx) = 2 secx
Tips for tackling such problems:
1) start on the most complicated side, it gives you more options to change things
2) it is USUALLY better to change everything to sines and cosines. these are the ones students are more familiar with, so it's easier to see connections
3) always consider working on both sides one at a time to bring them to the same thing. sometimes working on one side only to get the answer is too hard or even impossible in some sense

see below for solution