Trig Expression As An Algebraic Expression?

Question

Write sin(sin^-1 (x) + cos^-1 (y)) as an algebraic expression containing x and y (that is, without any trig functions).

HINT GIVEN IN BOOK:

Think of inverse trig functions.

Also: What is the easiest way or method for graphing inverse trig functions?

Answer 1

sin(sin^-1 (x) + cos^-1 (y))

let p=sin^-1 (x)=arcsinx then sinp=x
let q=cos^-1 (y)=arc cos y then cosq=y

sin(p+q)= sinpcosq+cospsinq

sin(sin^-1 (x) + cos^-1 (y))
=xy+sqrt(1-x^2) . sqrt(1-y^2)

Answer 2

The expression as you have written can be simplified using the trigonometric identity for the sine of a sum i.e. sin(A+B) = sin A cos B + cos A sin B and remembering that the inverse trigonometric functions are really asking the question “ determine the angle whose trig. function is a given value?”

Write sin(sin^-1 (x) + cos^-1 (y)) as an algebraic expression containing x and y (this expression can be thought of as sin(A+B))

Think of the sin^-1(x) as asking for the angle whose sine is x and cos^-1(y) as asking for the angle whose cosine is y. Next draw appropriate pictures for sin^-1(x) draw a right triangle in which you label one of the angles A
(i.e. A = sin^-1(x)). The side OPPOSITE this angle should be labeled x and the hypotenuse should be label 1. This means that the remaining side will be the square root of the quantity (1-x^2).

Draw another picture for the cos^-1 (y) and label one of the angles B (i.e. B =cos^-1 (y)). The side ADJACENT to this angle will be labeled y and the hypotenuse should be label 1. This now means that the remaining side will be square root of the quantity (1-y^2).

Using these 2 pictures and the trig. identity sin(A+B) = sin A cos B + cos A sin B you can now determine the requested algebraic expression.

See if this helps you get to a place where you can determine the answer yourself.

Answer 3

The best way to graph the inverse function
1) Graph the original function (sin x for example)
2) Turn it 90 degrees around the origin counter clockwise
3) Turn it around the y axis (imagine you put a mirror on the y axis)
4) You have a graph now consists of many identical segments around y axis. Eliminate all of them except the one closer to the origin.

Answer 4

let sin^-1(x) be a and cos^-1(y) be b
the expression will be Sin(a+b) andaccording to the rules of trigonmetry,this equal to Sin(a)Cos(b) + Sin(b)Cos(a)
Remember if Sin^-1(x) is a then Sin(a)=x
Cos^-1(y) is b then Cos(b)=y
Also if Sin(a)=x/1 according to pythogras Cos(b)=sqrt of (1-x^2)
Cos(b) =y/1 then Sin(b)=sqrt of (1-y^2)
now substitute: x y + [(1-y^2)(1-x^2)]^1/2.
I'll leave thte simplification of the expression to you,Otherwise that's my answer!