Shape of Sine Function

Theorem

The sine function is:

$(1): \quad$ strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$

$(2): \quad$ strictly decreasing on the interval $\closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$

$(3): \quad$ concave on the interval $\closedint 0 \pi$

$(4): \quad$ convex on the interval $\closedint \pi {2 \pi}$

Proof

From the discussion of Real Sine Function is Periodic, we have that:

$\sin \paren {x + \dfrac \pi 2} = \cos x$

The result then follows directly from the Shape of Cosine Function.

Graph of Sine Function

$\blacksquare$

Also see

• Properties of Real Sine Function

• Shape of Cosine 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