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Home Trigonometric Functions Derivatives of Trigonometric Functions Theorem 1: Continuity of Sine and Cosine
 
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Theorem 1: Continuity of Sine and Cosine

THEOREM 1

The functions x = cos θ and y = sin θ are continuous for all θ.

PROOF

We give the proof for θ in the first quadrant, 0 < θ < π/2. Let Δθ be infinitesimal and consider Figure 7.2.1.

07_trigonometric_functions-53.gif

Figure 7.2.1

Let Δs =07_trigonometric_functions-54.gifbe the length of the line PR.

Then

0 < Area of quadrilateral QPOR ≤ Area of sector POR,

0       <              ½Δs                    ≤             ½ Δθ.

Thus Δs is infinitesimal.

It follows that Δx and Δy are infinitesimal, whence the functions x = cos θ, y = sin θ are continuous.
Home Trigonometric Functions Derivatives of Trigonometric Functions Theorem 1: Continuity of Sine and Cosine

Last Update: 2010-11-25