Solutions of cos x equals cos a/Examples/cos 2 x equals sin 3 x
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Example of Use of Solutions of $\cos x = \sin a$
The equation
- $\cos 2 x = \sin 3 x$
has the general solution:
- $x = \paren {4 n + 1} \dfrac \pi {10}$
Proof
| \(\ds \cos 2 x\) | \(=\) | \(\ds \sin 3 x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {\dfrac \pi 2 - 3 x}\) | Cosine of Complement equals Sine | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 x\) | \(=\) | \(\ds 2 n \pi \pm \paren {\dfrac \pi 2 - 3 x}\) | Solutions of $\cos x = \cos a$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 x\) | \(=\) | \(\ds \begin {cases} 2 n \pi + \dfrac \pi 2 - 3 x \\ 2 n \pi - \dfrac \pi 2 + 3 x \end {cases}\) | simplifying | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 5 x\) | \(=\) | \(\ds \paren {4 n + 1} \dfrac \pi 2\) | simplifying | ||||||||||
| \(\, \ds \text {or} \, \) | \(\ds -x\) | \(=\) | \(\ds \paren {4 n - 1} \dfrac \pi 2\) | simplifying | ||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \paren {4 n + 1} \dfrac \pi {10}\) | ||||||||||
| \(\text {(2)}: \quad\) | \(\, \ds \text {or} \, \) | \(\ds x\) | \(=\) | \(\ds \paren {4 n + 1} \dfrac \pi 2\) | Cosine Function is Even and noting that $n$ is arbitrary | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \paren {4 n + 1} \dfrac \pi {10}\) | as equation $(1)$ subsumes equation $(2)$ |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Solution of equations: Example $1$.