Given an angle \(\displaystyle\theta\), the formulas for

encaderpwp

encaderpwp

Answered question

2022-03-25

Given an angle θ, the formulas for sin(θ2) and cos(θ2) are either positive or negative, which is why ± is used in their formulas. So, if tan(θ2)=sin(θ2)cos(θ2), and both the numerator and denominator are either positive or negative (and unless θ is defined, we must assume they can be both), shouldn't the formula for tan(θ2) also be ±?
Below are the equations I am referring to.
sin(u2)=±1cos(u)2
cos(u2)=±1+cos(u)2
tan(u2)=1cos(u)sin(u)=sin(u)1+cos(u)

Answer & Explanation

Cassius Villarreal

Cassius Villarreal

Beginner2022-03-26Added 11 answers

Basically, your first two equations involve square roots, and the third doesn't. The first requires a ± as sinu2 may be positive or negative, but by convention square roots are positive.
But (using duplication formulas for sine and cosine)
1cosusinu=1(12sin2(u2))2sin(u2)cos(u2)=2sin2(u2)2sin(u2)cos(u2)=sin(u2)cos(u2)=tan(u2)

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