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Section 3.3: Derivative of the Trigonometric Functions
Theory:
In this section we want to find d dx
sin(x) and d dx
cos(x), and the derivative of the other trigonometric functions: tan(x), cot(x), sec(x), csc(x). We first recall the addition formulas of sin(x) and cos(x). For numbers x, y we have sin(x + y) = sin(x) · cos(y) + sin(y) · cos(x), cos(x + y) = cos(x) · cos(y) − sin(x) · sin(y). Let us set up the limits to compute the derivatives of sin(x) and cos(x)
d sin(x) dx
= lim h→ 0
sin(x + h) − sin(x) h = lim h→ 0
sin(x) · cos(h) + sin(h) · cos(x) − sin(x) h [addition formula]
= lim h→ 0
[
sin(x) cos(h)^ −^1 h
]
and d cos(x) dx
= lim h→ 0
cos(x + h) − cos(x) h = lim h→ 0
cos(x) cos(h) − sin(x) sin(h) − cos(x) h = lim h→ 0
cos(x)(cos(h) − 1) − sin(x) sin(h) h [addition formula] = lim h→ 0
[
cos(x) cos(h h) −^1 − sin(x) sin( hh)
]
This computation tells us that we need to compute the following limits
hlim→ 0
sin(h) h and^ hlim→ 0
1 − cos(h) h.
1
Theorem 1
x^ lim→ 0
sin(x) x = 1 and^ xlim→ 0
1 − cos(x) x = 0
Proof. Proof of Theorem 1. Consider first the following picture:
sin(x) x tan(x)
cos(x)
Assume: Radius r = 1 Length of arc: x > 0, Area of section (pizza slice): 12 x (relation between arc and area of section): Indeed, note if x = 2π then area is A = π (full circle), if x = π then A = π/2 (half of circle), if x = π/2 then A = π/4 (quarter of circle), etc. Area of small triangle: 12 cos(x) sin(x), Area of large triangle: 12 tan(x). Since area of small triangle ≤ area of section ≤ area of large triangle: 1 2 sin(x) cos(x)^ ≤^
2 x^ ≤^
2 tan(x) =
sin(x) cos(x)
Multiplying by 2 and dividing by sin(x) leads to:
cos(x) ≤ x sin(x)
cos(x)
Taking reciprocals we obtain:
1 cos(x) ≥^
sin(x) x ≥^ cos(x).
cos(x) is continuous, thus limx→ 0 cos(x) = cos(0) = 1 and limx→ (^0) cos(^1 x) = 1. The Squeeze Theorem (for x < 0 we have same inequalities) implies there- fore:
xlim→ 0
sin(x) x = 1.
The derivative of the other trigonometric functions can be computed by using the rules of differentiation: d tan(x) dx
= d dx
sin(x) cos(x)
= cos(x) cos(x)^ −^ (−^ sin(x)) sin(x) cos^2 (x)
cos^2 (x)
= sec^2 (x),
d cot(x) dx
= d dx
cos(x) sin(x)
= (−^ sin(x)) sin(x)^ −^ cos(x) cos(x) sin^2 (x)
sin^2 (x)
= − csc^2 (x),
d sec(x) dx =^
d dx
cos(x) =^ −
− sin(x) cos^2 (x) = tan(x) sec(x), d csc(x) dx =^
d dx
sin(x) =^ −^
cos(x) sin^2 (x)
= − cot(x) csc(x).
Problems:
Problem 1. Compute the following limits
a) lim x→ 0
sin(3x) x
b) lim x→ 0 sin(5x) sin(7x)
c) lim x→ 0 tan
(^2) (5x) x^2
d) lim x→ 01 −^ cos(x) sin(2x)
- Problem 3. Find the tangent y = x^2 sin(x) to the point x = π
