Solutions of tan x equals tan a
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Theorem
Let $\alpha \in \R$ be fixed.
Let:
- $(1): \quad \tan x = \tan \alpha$
The solution set of $(1)$ is:
- $\set {x \in \R: \forall n \in \Z: x = n \pi + \alpha}$
Proof
From Tangent Function is Periodic on Reals:
- $\map \tan {\pi + x} = \tan x$
Hence:
\(\ds x\) | \(=\) | \(\ds n \pi + a\) |
$\blacksquare$
Examples
Solutions of $2 \sec^2 x = 5 \tan x$
The equation
- $2 \sec^2 x = 5 \tan x$
has the general solution:
- $\set {n \pi + \arctan \dfrac 1 2: n \in \Z} \cup \set {n \pi + \arctan 2: n \in \Z}$
where $\arctan$ denotes the (real) arctangent function.
Solutions of $\tan^2 x - 3 \tan x + 2 = 0$
Solutions of tan x equals tan a/Examples/tan squared x minus 3 tan x plus 2 equals 0
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(36)$