without a calculator, evaluate cos(9pi/4) PLEASE give an explanation, i know the answer already from using my calculator but im so confused of how to get it without one, which is sadly what my homework assignment is calling for.. please help! im so lost and i used to be so good at this last year:/
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The first thing for these type of problems is to bring the argument of the trig function into the range between 0 and 2pi, because the trig functions are periodic.
Obviously 9/4 = 2 1/4, so if we subtract 2pi (which doesn't change the result), we are left with
pi/4.
So... now you need the cosine of pi/4. In degrees, that's a 45 degree angle, for which the sin and cos must be the same (draw the unit circle and confirm this). But we also know that sin^2 x + cos^2 x = 1, so we know that 2 * cos^2 (pi/4) = 1
Which means that cos(pi/4) is either 1/sqrt(2) or -1/sqrt(2), and looking at the unit circle again tells us that it most be 1/sqrt(2).
Obviously 9/4 = 2 1/4, so if we subtract 2pi (which doesn't change the result), we are left with
pi/4.
So... now you need the cosine of pi/4. In degrees, that's a 45 degree angle, for which the sin and cos must be the same (draw the unit circle and confirm this). But we also know that sin^2 x + cos^2 x = 1, so we know that 2 * cos^2 (pi/4) = 1
Which means that cos(pi/4) is either 1/sqrt(2) or -1/sqrt(2), and looking at the unit circle again tells us that it most be 1/sqrt(2).