Proof of Derivatives of Trigonometric Functions Using Limit Definition
The derivative of trig functions proof including proof of the trig derivatives that includes sin, cos and tan. These three are actually the most useful derivatives in trigonometric functions. That being said, the three derivatives are as below:
d/dx sin(x) = cos(x)
d/dx cos(x) = −sin(x)
d/dx tan(x) = sec2(x)
Also, remember that the trigonometric functions sin(x) and cos(x) plays an important role in calculus. The AP Calculus however doesn't need knowing the proofs of these derivatives, but as long as a proof is accessible, there's always something to learn from it.
Derivative of Sin x Proof Using Algebraic Method
Proof of (d/dx) Sin(x): With the help of algebraic Method.
Given: \[\lim_{(d\rightarrow0)}\] sin(d)/d = 1.
Solve:
(d/dx) sin(x) = \[\lim_{(d\rightarrow0)}\] {sin(x+d) - sin(x)} / d
= lim {sin(x)cos(d} + cos(x)sin(d) - sin(x)) / d
= lim {sin(x)cos(d} - sin(x) )/d + lim cos(x)sin(d)/d
= sin(x) lim (cos(d) - 1)/d + cos(x) lim sin(d)/d
= sin(x) lim ( (cos(d)-1)(cos(d)+1))/(d(cos(d)+1)) + cos(x) lim sin(d)/d
= sin(x) lim (cos² (d)-1)/(d(cos(d)+1) + cos(x) lim sin(d)/d
= sin(x) lim -sin² (d)/(d (cos(d) + 1) + cos(x) lim sin(d)/d
= sin(x) lim (-sin(d)) × lim sin (d)/d × lim 1/ (cos(d)+1) + cos(x) lim sin(d)/d
= sin(x) × 0 × 1 × ½ + cos(x) × 1 = cos(x)
Derivative of Cos x Proof Using Algebraic Method
Proof of (d/dx) Cos(x): Taking the derivative of sine.
Derivative of cos(x) can be derived in the same way as (d/dx) sin(x) was derived or more simply from the outcome of (d - dx) sin(x).
Given: (d/dx) sin(x) = cos(x); [applying the Chain Rule.]
Solve:
cos(x) = sin(x + PI/2)
(d/dx) cos(x) = (d/dx) sin(x + PI/2)
= (d/du) sin (u) × (d-dx) (x + PI/2) (Set u = x + PI/2)
= cos(u) × 1 = cos(x + PI/2) = -sin(x)
Thus, derivatives of sin and cos are performed in a similar manner.
Derivative of Tan x Proof Using Algebraic Method
Derivative of Tan Proof of (d/dx) Tan(x): Taking the derivatives of sine and cosine.
Given: (d/dx) sin(x) = cos(x); (d/dx) cos(x) = -sin(x); [applying the Quotient Rule]
Solve:
Tan(x) = sin(x) / cos(x)
(d/dx) tan(x) = (d/dx) sin(x)/cos(x)
= (cos(x) (d/dx) sin(x) - sin(x) (d/dx) cos(x))/cos²(x)
= (cos(x)cos(x) + sin(x)sin(x))/cos²(x)
= 1 + tan²(x) = sec²(x)
Solved Examples
Now that we are known of the derivative of sin, cos, tan, let's learn to solve the problems associated with derivative of trig functions proof.
Example:
Determine the derivative of:
f(x) = x² sin(3x)
Solution
We will apply the chain and the product rules.
F '(x) = (2x)(sin(3x)) + (x²)(3cos(3x))
= 2x sin (3x) + 3x² cos (3x)
Example:
The amount A of food available at day (d) of the year for an Indian bear can be modelled by the equation
A(d) = 320 cos 2 π(d - 120)/365
Find out on what day does the bear have the greatest food available?
Solution:
You are asked to find a maximum, which includes taking the 1st derivative and establishing it equivalent to zero. We have
A'(d) = -320(2 π) 2p (d - 120)/365
Establishing it equivalent to zero and solving provides us.
2p (d - 120)/365 = kπ
When k = 0, we obtain d = 120 and when k = 1, we obtain d = 302.5. The 2nd derivative test will tell us that d = 120, provides us with a maximum and d = 320 provides us with a minimum. Other values of k will provide us the same day of the year. We thus come to the conclusion that the bear has the most food supply on April 30.
Fun Facts
Angles in trigonometric functions will always be given in radians.
Calculus and Degrees never go together.
If you ever hear the word "Degree" in trig functions, the appropriate question to ask would be "Do you mean Celsius or Fahrenheit"?
FAQs on Derivatives Trig Proof with Step by Step Explanations
1. What is the derivative of sin x and how is it proved?
The derivative of sin x is cos x, proved using the limit definition of a derivative. Using first principles:
- Start with: d/dx (sin x) = limh→0 [sin(x + h) − sin x] / h
- Apply the identity: sin(x + h) = sin x cos h + cos x sin h
- Simplify and use limits: limh→0 (sin h / h) = 1 and limh→0 (cos h − 1) / h = 0
2. How do you prove the derivative of cos x?
The derivative of cos x is −sin x, proved using the limit definition and trig identities. Using first principles:
- Start with: d/dx (cos x) = limh→0 [cos(x + h) − cos x] / h
- Use identity: cos(x + h) = cos x cos h − sin x sin h
- Apply limits: limh→0 (sin h / h) = 1 and limh→0 (cos h − 1) / h = 0
3. What is the derivative of tan x and how is it derived?
The derivative of tan x is sec² x, derived using the quotient rule or known trig derivatives. Since tan x = sin x / cos x:
- Apply the quotient rule: (u/v)' = (u'v − uv') / v²
- Differentiate sin x and cos x
- Simplify using sin²x + cos²x = 1
4. Why must angles be in radians when proving trigonometric derivatives?
Angles must be in radians because the key limit limh→0 (sin h / h) = 1 only holds in radians. If degrees are used, this limit is not equal to 1, and the derivative formulas become incorrect. All standard trigonometric derivative proofs, including those for sin x and cos x, assume the angle is measured in radians.
5. What is the limit sin h / h as h approaches 0?
The limit limh→0 (sin h / h) equals 1 when h is measured in radians. This fundamental trigonometric limit is essential for proving derivatives of sin x and cos x. It is usually proved using the squeeze theorem and unit circle geometry, and it forms the foundation of derivative trig proofs.
6. How do you prove trigonometric derivatives using first principles?
Trigonometric derivatives are proved using the limit definition of derivative and trig identities. The steps are:
- Use f'(x) = limh→0 [f(x + h) − f(x)] / h
- Apply appropriate angle addition identities
- Separate terms and factor
- Use standard limits such as sin h / h = 1 and (cos h − 1)/h = 0
7. What are the basic trigonometric derivative formulas?
The basic trigonometric derivative formulas are standard results used in calculus. The main ones are:
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec² x
- d/dx (cot x) = −csc² x
- d/dx (sec x) = sec x tan x
- d/dx (csc x) = −csc x cot x
8. Can you give an example of a trigonometric derivative proof?
Yes, for example, to prove the derivative of sin x at x = 0, we use first principles. Start with:
- f'(0) = limh→0 (sin h − 0)/h
- This becomes limh→0 (sin h / h)
9. What common mistakes occur in derivative trig proofs?
The most common mistake in trigonometric derivative proofs is using degrees instead of radians. Other common errors include:
- Incorrect use of angle addition identities
- Forgetting key limits like sin h / h = 1
- Algebraic simplification mistakes
10. What is the difference between proving and applying trigonometric derivatives?
Proving trigonometric derivatives uses the limit definition, while applying them uses the known derivative formulas directly. In proofs:
- You start from first principles
- You use trig identities and fundamental limits
- You directly use results like d/dx (sin x) = cos x
- You may combine them with product, quotient, or chain rules
