Question

Evaluate sin `[tan^-1 tan((3pi)/4)]`.

Answer 1

Step 1: Simplify the inner tangent expression

We use the trigonometric identity tan (π − θ) = −tanθ to bring the angle into the principal range:

`tan ((3pi)/4)`

= `tan (pi - pi/4)`

= `- tan (pi/4)`

tan `(-pi/4)`

Step 2: Evaluate the inverse tangent

Now, substitute the simplified value back into the inverse function:

`tan^-1 (tan (-pi/4)) = -pi/4`

Step 3: Solve for Sine

`sin (-pi/4)`

= −sin`(pi/4)`

= `-1/sqrt2 or -sqrt2/2`