# Given the trigonometric value of the double angle, how to find the trigonometric value taught by double？

Known this

Tan[2 A] == 2 Sqrt[2]


How do you evaluate its value?

Tan[A] ==?   Sin[A]==?   Cos[A]==?

• You can do slnt = Solve[Tan[2 A] == 2 Sqrt[2], A] /. C[1] -> 0 and then Tan[A] /. slnt // FullSimplify to get $\frac{1}{\sqrt{2}}$. You can also leave C[1] and still do the same
– bmf
Commented Feb 16, 2023 at 2:12
• Does Solve[{2 Sqrt[2] == TrigExpand[Tan[2 a]], ta Cos[a] == Sin[a]}, ta, {Cos[a], Sin[a]}] do what you were expecting? Commented Feb 16, 2023 at 2:14
• @bmf leave C[1] the result is: ConditionalExpression[ Tan[1/2 (ArcTan[ 2 Sqrt[2]] + \[Pi] ConditionalExpression[1, \[Placeholder]])], ConditionalExpression[1, \[Placeholder]] \[Element] Integers]} Commented Feb 16, 2023 at 2:29
• @csn899 ok, I still don't understand the issue
– bmf
Commented Feb 16, 2023 at 2:35
• Yes, so you need to recall sign patterns for trigonometric functions, or at least recall that tan > 0 iff a > 0. Commented Feb 16, 2023 at 12:11

I think I understand what is wanted.

sltn = Solve[Tan[2 A] == 2 Sqrt[2], A] // Flatten // Normal;
res = Tan[A] /. sltn;
rule = Tan[a_ + b_] :> (Tan[a] + Tan[b])/(1 - Tan[a] Tan[b]);


and then

FullSimplify /@ ((Expand /@ res) /. rule) // FullSimplify


So, essentially the trick is to implement the rule of the Tan of the sum. Likewise for the rest.

• This is powerful and derives the doubly angle formula that represents tangents in terms of tangents. How do I limit the scope of A? I tried to run an error in the latter plus range. Another question, why can't the software directly use the Trigexpand command to generate a doubly angle formula that represents tangent with tangent? Commented Feb 16, 2023 at 5:11
• TrigExpand[Tan[2 a]]==(2 Cos[a] Sin[a])/(Cos[a]^2 - Sin[a]^2) Commented Feb 16, 2023 at 5:14
• @csn899 related to your second question: from the documentation TrigExpand splits up sums and integer multiples that appear in arguments of trigonometric functions, and then expands out products of trigonometric functions into sums of powers, using trigonometric identities when possible. Which means that TrigExpand automatically simplifies even further to Sin and Cos. That is my understanding of the matter. I don't understand what you mean by limit the scope of A
– bmf
Commented Feb 16, 2023 at 5:22
• For example, the scope of Limit A is:0<=A<=pi/2 Commented Feb 16, 2023 at 8:15
• @csn899 You can add assumptions to Solve like this: sltn = Assuming[0 <= A < 2 Pi, Solve[Tan[2 A] == 2 Sqrt[2], A]] //Flatten // Normal essentially this fixes the values of C[1] so you can do it at the end to be honest
– bmf
Commented Feb 16, 2023 at 8:21