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Use the trigonometric subtraction formula for sine to verify this identity: `sin(pi/2 -...

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atrinity | (Level 1) Honors

Posted April 20, 2013 at 11:20 PM via web

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Use the trigonometric subtraction formula for sine to verify this identity:

`sin(pi/2 - x) =cosx`

 

Please provide a reason with the 3 steps.

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mjripalda | High School Teacher | (Level 1) Senior Educator

Posted April 21, 2013 at 12:42 AM (Answer #1)

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`sin(pi/2 -x)=cosx`

To prove, at the left side, apply the identity for difference of two angles of sine which is `sin(A - B) = sinAcosB - sinBcosA` .

`sin(pi/2)cosx-sinxcos(pi/2)`

Since `sin (pi/2)=1` and `cos(pi/2)=0` , plug-in these values to the equation.

`1*cosx - sinx *0 = cosx`

`cosx - 0 = cos x`

`cosx=cos x`    (True)

Since simplifying left side results to a true condition, this proves that the given equation is an identity.

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violy | High School Teacher | (Level 3) Assistant Educator

Posted April 21, 2013 at 12:55 AM (Answer #2)

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Use the difference between two angles for sine. 

`sin(a - b) = sinacosb - cosasinb`

So, we will have:

`sin(pi/2 - x) = sin(pi/2)cosx - cos(pi/2)sinx`

We know that sin(pi/2) = 1, and cos(pi/2) = 0.

Plug-in those values.

`sin(pi/2)cosx - cos(pi/2)sinx = 1*cosx - 0*sinx`

Simplify. 

`1*cosx - 0*sinx = cosx`

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