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Section 4.7
Can you do (a) as a guidefor me to do the rest?
Thanks.
Can you do (a) as a guidefor me to do the rest?
Thanks.
I did d), leaving a) to you
Let sin^-1(x)=θ
=>x=sin(θ)=cos(π/2-θ)
=>cos^-1(x)=π/2-θ=π/2-sin^-1(x)
=>sin^-1(x)+cos^-1(x)=π/2
sin^-1(x) is equal to the angle (for example theta) whose sine is x:
For part (a) manipulate one side and prove it is equal to the other side
hint: let sin^-1(-x) be equal to y
y=sin^-1(x)
I will play with part (a) later on my break or tomorrow morning.
Prove arcsin (-x) =-arcsin x
Let u = arcsin (- x) and v = - arcsin( x)
Then, sin (u) = - x and -v = arcsin(x) =>x = sin(-v) = - sin( v)
or x = - sin( u )
Comparing the two, x = - sin (u) = - sin (v) => u = v
Therefore, arcsin (-x) = -arcsin(x)
c.
prove: arctan(x)+arctan(1/x)=pi/2, x>0
Consider the point P=(1,x), in the first quadrant, with corresponding angle 0< α <π/2.
Let β=π/2-θ.
Then, also, 0<β <π/2 and tan(β)=tan(π/2-α )=cot(α )=1/tan(α )=1/x
It follows that arctan(x)+arctan(1/x)=α + β=π/2