help w/ simple trig equation

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help w/ simple trig equation

Postby cjcrawford on Tue Mar 18, 2014 4:28 pm

Hi - out of a text the following example is given:

if sin2x = -cos(-x + 9) then find x. (in degrees)

1. sin2x = -cos(-x + 9) = cos(-x +9 +180)
2. but 2x - x + 9 + 180 = 90
3. x = -99

steps 1 and 3 are clear to me but I'm missing why in step 3 we can add the angles to 90. All insight appreciated. Thanks.

Chris
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Re: help w/ simple trig equation

Postby anonmeans on Tue Mar 18, 2014 7:40 pm

cjcrawford wrote:if sin2x = -cos(-x + 9) then find x. (in degrees)

1. sin2x = -cos(-x + 9) = cos(-x +9 +180)

I think they're using that cos(A) = -cos(A + 180).

cjcrawford wrote:2. but 2x - x + 9 + 180 = 90

I think maybe they're using that cos(A) = sin(A) for A = 45, so they're equal for A = 45, so A + A = 2A = 2(45) = 90? And they added the 180 to make the cos positive so you can use that cos(A) = sin(A) for A = 45.
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Re: help w/ simple trig equation

Postby cjcrawford on Tue Mar 18, 2014 9:06 pm

That's what I thought at first:

sinx = cosy only at 45 degrees therefore
x + y = 45 +45 = 90
so 2x - x + 9 + 180 = 90

but...

by the same token that would mean that for sin() to equal cos(), each angle must be 45 so
2x = 45 and
(-x + 9 +180) = 45
if you solve for this you get x = 22.5 in the first case and 144 in the second which does NOT work. Still confused!

Chris
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Re: help w/ simple trig equation

Postby nona.m.nona on Wed Mar 19, 2014 1:37 pm

First, confirm the proposed solution:

sin(2x) for x = -99 is sin(2(-99)) = sin(-198) = sin(360 - 198) = sin(162) = sin(180 - 162) = sin(18)
-cos(-x + 9) for x = -99 is -cos(-9 + -99) = -cos(-108) = -cos(360 - 108) = -cos(252) = -sin(252 + 90) = -sin(342) = -sin(342 - 360) = -sin(-18) = sin(18)

So the solution is correct, but not because the angle measures are equal to 45 degrees.

Let 2x = A and -x + 9 + 180 = B. Then we have sin(A) = cos(B). One of the phase-shift identities states that cos(B) = sin(90 - B), so A = 90 - B. Adding these, we get A + B = (90 - B) + B = 90. The result follows.
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