Question 1203961
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List of common trig identities
<a href = "https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf">https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf</a>


The identity we'll be using is 
{{{tan(pi/2 - x) = cot(x)}}}
it is found at the bottom of page 2 under the subsection "Cofunction Formulas".


Cotangent is the reciprocal of tangent, meaning that if tan(x) = 4/5, then cot(x) = 5/4.
This then leads back to {{{tan(pi/2 - x) = 5/4}}}


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Here is another approach.


Draw out a right triangle with these properties:
opposite = 4
adjacent = 5
theta = reference angle


We are then effectively given that tan(theta) = 4/5
The expression {{{pi/2 - theta}}} represents the other acute angle of this right triangle.
This is because the two acute angles add to pi/2 radians (aka 90 degrees). The acute angles are complementary.


With reference to angle {{{pi/2 - theta}}}, the positions of "opposite" and "adjacent" swap places.
This will mean that we have:
opposite = 5
adjacent = 4
pi/2 - theta = reference angle


Therefore, tan(pi/2 - theta) = opposite/adjacent = <font color=red>5/4</font>


Here is a diagram of what I refer to
{{{
drawing(400,400,-5,5,-5,5,
line(-3,-3,3,3),
line(3,3,3,-3),
line(3,-3,-3,-3),
line(3-0.5,-3,3-0.5,-3+0.5),
line(3-0.5,-3+0.5,3,-3+0.5),
locate(3.5,0,"4"),
locate(0,-3.2,"5"),
locate(3-1.2,3-1.2,pi/2-theta),
locate(-3+0.8,-3+0.5,theta),
locate(-4.5,-4,matrix(1,2,"Diagram","not")),
locate(-4.5,-4.5,matrix(1,2,"to","scale"))
)
}}}
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