SOLUTION: How do I solve the equation the equation sin(theta) - cos(theta) = sqrt( 2 ) ?

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Question 1058119: How do I solve the equation the equation sin(theta) - cos(theta) = sqrt( 2 ) ?
Answer by ikleyn(52835) About Me  (Show Source):
You can put this solution on YOUR website!
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How do I solve the equation the equation sin(theta) - cos(theta) = sqrt( 2 ) ?
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sin%28theta%29+-+cos%28theta%29 = sqrt%282%29.

Multiply both sides by sqrt%282%29%2F2. You will get

%28sqrt%282%29%2F2%29%2Asin%28theta%29+-+%28sqrt%282%29%2F2%29%2Acos%28theta%29 = 1.             (1)

Notice and use that sqrt%282%29%2F2 = cos%28pi%2F4%29 = sin%28pi%2F4%29.

Then from (*) you will get 

cos%28pi%2F4%29%2Asin%28theta%29+-+sin%28pi%2F4%29%2Acos%28theta%29 = 1.       (2)

Now use the formula sin(a)*cos(b) - cos(a)*sin(b) = sin(a-b). Then from (2) you will get

-sin%28pi%2F4-theta%29 = 1.

It implies 

pi%2F4+-+theta = 3pi%2F2.

Then theta = pi%2F4+-+3pi%2F2 = -5pi%2F4.

It is the same as theta = 3pi%2F4 in the interval 0 <= theta < 2pi.

The plot below confirms the solution ( 3pi%2F4 ~= 2.33 )



Plots y = sin%28theta%29+-+cos%28theta%29 and y = sqrt%282%29

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    - Solving typical problems on trigonometric equations
    - Solving more complicated problems on trigonometric equations
    - Solving advanced problems on trigonometric equations
    - OVERVIEW of lessons on calculating trig functions and solving trig equations