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Question

Exact value of:
sin(arctan(-1)) (on the interval [0,2pi].)
sin=(-pi/4)=(-sqrt2/2)

Is this correct? I'm so confused.

Answer 1

OK, first let's try to figure out the arctan part. Arctan(-1) means the angle whose tangent is -1, right? And the tangent is the sine divided by the cosine. So you're looking for angles whose sines and cosines are equal, but with opposite signs. They point out to the right and down, or out to the left and up, at 45-degree angles. That means the angles are 135 degrees and 315 degrees, or 3pi/4 and 7pi/4 radians.

Now, the question asked you to find the sines of these two angles. The sine of 3pi/4 is sqrt(2)/2, and the sine of 7pi/4 is -sqrt(2)/2. So your answer is +/- sqrt(2)/2.

Answer 2

Amy has given a great verbal description of the solution.

Mathematically it would be written like so:

Let θ = arctan(-1)
So tanθ = -1

So θ is an angle that lies either in the 2nd or the 4th quadrant.
For 0° ≤ θ ≤ 360° this mean that θ lies between 90° and 180° or between 270° and 360°.

In radian measure we would say this statement as:

For 0 ≤ θ ≤ 2π this mean that θ lies between π/2 and π or between 3π/2 and 2π.

That means θ = 180° - 45° (π - π/4) or 360° - 45° (2π - π/4)
= 135° (3π/4) or 315° (7π/4)

Then sinθ = sin135° (= sin3π/4) or sin315° (= sin7π/4)
= 1/√2 or -1/√2 (Remembering that sinθ >0 in 2nd quadrant and < 0 in 4th quadrant)

So sin (arctan (-1)) = ±1/√2 as stated above.