How would you find the exact value of the six trigonometric function of 3π/4?

1 Answer
Dean R.
May 8, 2018

cos({3pi}/4)= - 1/sqrt{2}

sin({3pi}/4)= 1/sqrt{2}

tan({3pi}/4) = -1

sec({3pi}/4) = - sqrt{2}

csc({3pi}/4)= sqrt{2}

cot({3pi}/4)= -1

Explanation:

[3 pi}/4 is 135^circ = 3 times 45^circ

Of course 30/60/90 and 45/45/90 are the only two triangles students are expected to know "exactly." We have to know the multiples of these triangles in the other quadrant. Here we have 45/45/90 in the second quadrant.

We start from cos(45^circ)=sin(45^circ)=1/sqrt{2}. The angle 135^circ is supplementary to 45^circ, so has the opposite cosine and the same sine.

cos({3pi}/4)= - cos(pi - {3pi}/4)=- cos(pi/4)= - 1/sqrt{2}

sin({3pi}/4)=sin( pi - {3pi}/4) = sin(pi/4) = 1/sqrt{2}

tan({3pi}/4) = {sin({3pi}/4)}/{cos({3pi}/4)} = {1/sqrt{2}}/{-1/sqrt{2}}=-1

sec({3pi}/4)=1/cos({3pi}/4) = - sqrt{2}

csc({3pi}/4)=1/sin({3pi}/4) = sqrt{2}

cot({3pi}/4)=1/tan({3pi}/4) = -1

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