# Trigonometry question

1. Jan 17, 2007

### John O' Meara

The voltage E applied to the sending end of a high-pressure transmitting line is connected to the voltage e at the receiving end by the equation
E^2=(e*cos(x) + a)^2 + (e*sin(x) + b)^2, where a and b are constants. Expand the right-hand side of the equation and by expressing a*cos(x) + b*sin(x) in the form R*cos(x + alpha) show that the maximum and minimum values of R, as x varies, are e +/-sqr(a^2 + b^2)? On expanding I get the following:

E^2 = e^2*((cos(x))^2 + (sin(x))^2) + a^2 + b^2 + 2*e*(a*cos(x) + b*sin(x))
E^2 = e^2 + a^2 + b^2 + 2*e*sqr(a^2 + b^2)*(cos(alpha)*cos(x) + sin(alpha)*sin(x)); where,
cos(alpha) = a/sqr(a^2 + b^2), and sin(alpha) = b/sqr(a^2 + b^2), and tan(alpha) = b/a. Therefore
E^2 = e^2 + a^2 + b^2 +2*e*sqr(a^2 + b^2)(cos(x - alpha)), so I get R = 2*e*sqr(a^2 + b^2) not what it is claimed above. What is the next step? Many thanks.

2. Jan 18, 2007

### chanvincent

Let me tidy up your equation.. otherwise no one is able to read that.....

Okay, I've finished my part... someone gonna help him.....

Last edited: Jan 18, 2007
3. Jan 18, 2007

### chanvincent

I believe you mis-understand what the question is asking......

The question is asking you to express $$acos(x) + bsin(x)$$ in term of $$Rcos(x+\alpha)$$, then show the maximum and minimum value of E is $$e+/-\sqrt{a^2+b^2}$$, That would be more make sense....

Last edited: Jan 18, 2007
4. Jan 19, 2007

### John O' Meara

Thanks for taking the time to make my post readable chanvincent.

5. Jan 19, 2007

### John O' Meara

Expanding $$a cos(x) + b sin(x) [/]tex] I get : [tex] = \sqrt{a^2 + b^2}(a/\sqrt{a^2 + b^2}cos(x) + b/\sqrt{a^2 + b^2}sin(x)) = \sqrt{a^2 + b^2}(sin(\alpha)cos(x) + cos(\alpha)sin(x))$$, where $$\(alpha)=tan(a/b)$$, this expands out to: $$\sqrt{a^2 + b ^2}(1/2( sin(\(alpha) + x) + sin(\alpha - x)) + 1/2( sin(\alpha + x) - sin(\alpha -x)))$$
, which reduces to $$\sqrt{a^2 + b^2}(sin(\alpha + x))$$.The question is how do you expand it to get [tex] R cos(x + \alpha)[\tex], and what do you do then.

(Edited by HallsofIvy to fix tex. John, you put [ \tex ] when you should have [ /tex ] to end the tex.)

Last edited by a moderator: Jan 19, 2007