SOLUTION: If {{{ tan(x) = 4 / 5 }}} then, without a calculator, find {{{tan(pi / 2 - x) }}}. Write your answer as a fraction.

Algebra ->  Trigonometry-basics -> SOLUTION: If {{{ tan(x) = 4 / 5 }}} then, without a calculator, find {{{tan(pi / 2 - x) }}}. Write your answer as a fraction.      Log On


   

Question 1203961: If +tan%28x%29+=+4+%2F+5+ then, without a calculator, find tan%28pi+%2F+2+-+x%29+. Write your answer as a fraction.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

List of common trig identities
https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

The identity we'll be using is
tan%28pi%2F2+-+x%29+=+cot%28x%29
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%28pi%2F2+-+x%29+=+5%2F4

---------------------------------------------------

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%2F2+-+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%2F2+-+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 = 5/4

Here is a diagram of what I refer to