Cosine of Zero is One

Theorem

$\cos 0 = 1$

where $\cos$ denotes the cosine.

Proof

Recall the definition of the cosine function:

$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$

Thus:

$\cos 0 = 1 - \dfrac {0^2} {2!} + \dfrac {0^4} {4!} - \cdots = 1$

$\blacksquare$

Also see

• Sine of Zero is Zero
• Tangent of Zero
• Cotangent of Zero
• Secant of Zero
• Cosecant of Zero

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Special angles
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.3 \ (1) \ \text{(i)}$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Trigonometric values for some special angles
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Trigonometric values for some special angles