Discover the Solution to Sin(arctan(x/4)) with Expert Guidance

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Homework Help Overview

The discussion revolves around the expression Sin(arctan(x/4)), with participants exploring the relationship between trigonometric functions and right triangles. The subject area includes trigonometry and the properties of inverse functions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants suggest using a right triangle to visualize the problem, specifically setting up a triangle with sides corresponding to the tangent ratio. There are attempts to derive the sine value from the triangle's dimensions, and questions arise about the correctness of the derived expression.

Discussion Status

The discussion includes various approaches to visualize and calculate the sine of the angle derived from the arctangent function. Some participants provide guidance on drawing the triangle and applying the Pythagorean theorem, while others express uncertainty about the correctness of their calculations.

Contextual Notes

Participants are working under the constraints of a homework context, and there is a noted lack of consensus on the final expression derived for Sin(A). The original poster indicates a long absence from math, which may affect their confidence in the discussion.

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[Mentor's note: This thread was originally posted in a non-homework forum, so it doesn't follow the homework template.]

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Sin(arctan(x/4))= ?

Been over 2 years since I've done some math, a little help please?
 
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Set up a right triangle with sides x and 4, so that the tangent of one of the angles is x/4, i.e., tanθ=x/4. Then θ=tan^{-1}(x/4) . From the drawing, figure out the value of sinθ.
 
It might help you if you draw a triangle and split the expression above into component parts.

First, how would you triangle look if you were to show what arctan(x/4) meant?
 
Ok I:

Drew a triangle in quadrant 1 to represent x/4 and labeled the angle A for random sake
then used Pythagorean theorem to find the hyp
after solving for sign and rationalizing I came up with:

SinA= (x(√(x^2)+16)/((x^2)+16) ------ √ ending after the first 16
sound right?
 
Yes, ##\sin(A)=x\frac{\sqrt{x^2+16}}{x^2+16}##.

ehild
 

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