How do you find the exact trigonometric ratios for the angle 3pi/4?

Question

I took precalculus two years ago, and I don't remember where to start with this type of problem. Can someone please guide me on how to find the lengths of the sides of the triangle and then calculate the trigonometric ratios for the angle ( \frac{3\pi}{4} )? I would appreciate a thorough explanation, as I am a bit rusty with this material. Thank you!

Braden Dicki · Accepted Answer

With every angle there is associated a reference angle, which is the positive acute angle that the terminal side of the given angle makes with the (positive or negative) x-axis. An angle and its reference angle have the same value for their trig functions, except possibly for sign.

3π/4 is in the second quadrant, so its cosine is negative and its sine is positive. From these you can determine the sign of the other four trig functions.

The reference angle for 3π/4 is π/4. This is an angle for which you are expected to know at least the sine and cosine: sin(π/4) = cos(π/4) = √2 / 2. Therefore,

sin(3π/4) = √2 / 2

cos(3π/4) = -√2 / 2

tan(3π/4) = sin(3π/4)/cos(3π/4) = -1

csc(3π/4) = 1/sin(3π/4) = √2

sec(3π/4) = 1/cos(3π/4) = -√2

cot(3π/4) = 1/tan(3π/4) = -1

Anonymous · Answer

How To Find Trigonometric Ratios

Emmet Cole

How do you find the exact trigonometric ratios for the angle 3pi/4?

2 Answers

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