# Math Help - trignometric equations help

1. ## trignometric equations help

(b) The diagram shows part of the curve with equation y=cos(px−q)° , where p and q are positive constants and q<180. The curve cuts the x-axis at points A, B and C, as shown.
Given that the coordinates of A and B are (100, 0) and (220, 0) respectively:
(i) Write down the coordinates of C.
(ii) Find the value of p and the value of q.
I have guessed that the coordinates of C is (340,0), which is correct, but is there any 'formal' way of working out the coordinates of C?

Also can someone show me how to work out the vales of p and q?

thank you.

2. Hi

$\cos (px - q) = 0 \implies px - q = 90 + 180 k$

A, B and C are consecutive zeros of cos(px-q) therefore

$px_A - q = 90 + 180 k$

$px_B - q = 90 + 180 (k+1)$

$px_C - q = 90 + 180 (k+2)$

Subtracting gives
$p(x_B - x_A) = 180$
$p(x_C - x_B) = 180$

Then $p(x_C - x_B) = p(x_B - x_A)$ or $x_C = 2x_B - x_A = 340$

And $p = \frac{180}{x_B - x_A} = \frac32$