Find the cosine tangent of the cotangent if sine = 24/25.

| May 2, 2021 | education

In the task for a certain angle α, the value of the sine is given, that is, sin α = 24/25, however, there is no information about where (in which coordinate quarter) this angle α is located. The fact is that, depending on the coordinate quarter where the angle is located, different trigonometric functions can take positive or negative values. For example, the sine function has positive values ​​in both I and II coordinate quarters, while all other functions: in the I coordinate quarter take positive values, and in the II coordinate quarter – negative values. To fulfill the requirement of the task, we will use the formulas: tgα = sinα / cosα, ctgα = cosα / sinα and sin2α + cos2α = 1 (basic trigonometric identity), which we will rewrite as cosα = ± √ (1 – sin2α).
Consider 2 cases: a) 0 <α <π / 2, that is, α belongs to the I coordinate quarter; b) π / 2 <α <π, that is, α belongs to the II coordinate quarter.
In case a) we have: cosα = + √ (1 – (24/25) ^ 2) = + √ ((625 – 576) / 625) = √ (49/625) = 7/25, therefore, tgα = ( 24/25) / (7/25) = 24/7 and ctgα = (7/25) / (24/25) = 7/24.
Similarly, in case b) we have: cosα = -√ (1 – (24/25) ^ 2) = -7/25, therefore, tgα = (24/25) / (-7/25) = -24/7 and ctgα = (-7/25) / (24/25) = -7/24.



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