Apparent trigonometric inconsistencies

[intervoxel]

a is the opposite side
b is the adjacent side
c is the hypotenuse.
x is the angle

Problem: expression for sin x as a function of a and c.

Solution:

Identities taken from textbooks:

sin x = a/b (1)
tan x = a/c (2)
sin x = sqrt(tan^2 x / (1 + tan^2 x)) (3)

substituting (2) in (3), we have

sin x = a / sqrt(a^2 + c^2) (4)

On the other hand,

a^2 + b^2 = c^2 => b=sqrt(c^2 - a^2)

substituting this in (1), we have

sin x = a / sqrt(-a^2 + c^2) (5)

(4) != (5) ? how come?

What's wrong here, please?

 

[SteamKing]

a is the opposite side
b is the adjacent side
c is the hypotenuse.
x is the angle

Problem: expression for sin x as a function of a and c.

Solution:

Identities taken from textbooks:

sin x = a/b (1)
tan x = a/c (2)

You don't cite a source for (1) and (2) above, but each is incorrect. The sine is the opposite over the hypotenuse and the tangent is the opposite over the adjacent.

Here is a nifty graphic:

 

[intervoxel]

Oops!
Thank you for the prompt answer.

 

[DaveC426913]

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