Shape of Cosine Function
Theorem
The cosine function is:
- $(1): \quad$ strictly decreasing on the interval $\closedint 0 \pi$
- $(2): \quad$ strictly increasing on the interval $\closedint \pi {2 \pi}$
- $(3): \quad$ concave on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
- $(4): \quad$ convex on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$
Proof
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From the discussion of Real Cosine Function is Periodic, we know that:
- $\cos x \ge 0$ on the closed interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$
and:
- $\cos x > 0$ on the open interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$
From the same discussion, we have that:
- $\map \sin {x + \dfrac \pi 2} = \cos x$
So immediately we have that $\sin x \ge 0$ on the closed interval $\closedint 0 \pi$, $\sin x > 0$ on the open interval $\openint 0 \pi$.
But $\map {\dfrac \d {\d x} } {\cos x} = -\sin x$ from Derivative of Cosine Function.
Thus from Derivative of Monotone Function, $\cos x$ is strictly decreasing on $\closedint 0 \pi$.
From Derivative of Sine Function it follows that:
- $\map {\dfrac {\d^2} {\d x^2} } {\cos x} = -\cos x$
On $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$ where $\cos x \ge 0$, therefore, $\map {\dfrac {\d^2} {\d x^2} } {\cos x} \le 0$.
From Second Derivative of Concave Real Function is Non-Positive it follows that $\cos x$ is concave on $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
The rest of the result follows similarly.
$\blacksquare$
Graph of Cosine Function
Also see
- Shape of Sine Function
- Shape of Tangent Function
- Shape of Cotangent Function
- Shape of Secant Function
- Shape of Cosecant Function
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Signs and Variations of Trigonometric Functions
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.5 \ (1)$