Algebraic Methods and Example Problems for Trig Derivatives
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 of Trigonometric Functions with Proofs
1. What are the standard derivatives of the six trigonometric functions?
The derivatives of the six basic trigonometric functions are fundamental in calculus. They 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
These results are proven using methods like the first principle and the quotient rule.
2. How is the derivative of sin(x) proven using the first principle?
The proof for the derivative of sin(x) uses the limit definition of a derivative, also known as the first principle. The key steps involve using the sine addition formula and two fundamental trigonometric limits.
- Start with the definition: f'(x) = lim(h→0) [f(x+h) - f(x)] / h.
- Substitute f(x) = sin(x): d/dx(sin x) = lim(h→0) [sin(x+h) - sin(x)] / h.
- Apply the angle addition formula sin(A+B): lim(h→0) [sin(x)cos(h) + cos(x)sin(h) - sin(x)] / h.
- Rearrange terms: lim(h→0) [cos(x)sin(h)/h - sin(x)(1-cos(h))/h].
- Use the known limits: lim(h→0) sin(h)/h = 1 and lim(h→0) (1-cos(h))/h = 0.
- This simplifies to cos(x) * (1) - sin(x) * (0), which equals cos(x).
3. How are the derivatives for tan(x) and sec(x) proven using the quotient rule?
Once the derivatives for sin(x) and cos(x) are known, derivatives for other trig functions can be proven using the quotient rule. To find the derivative of tan(x):
- Express tan(x) as sin(x)/cos(x).
- Apply the quotient rule, d/dx [u/v] = [v(du/dx) - u(dv/dx)] / v², where u = sin(x) and v = cos(x).
- Substituting the derivatives gives: [cos(x) * cos(x) - sin(x) * (-sin(x))] / cos²(x).
- This simplifies to (cos²(x) + sin²(x)) / cos²(x).
- Using the identity sin²(x) + cos²(x) = 1, the result is 1/cos²(x), which equals sec²(x).
A similar method is used for sec(x) by writing it as 1/cos(x).
4. What is the geometric meaning behind the limit lim(θ→0) sin(θ)/θ = 1?
The limit lim(θ→0) sin(θ)/θ = 1 has a profound geometric meaning related to the unit circle. It signifies that for a very small angle θ (measured in radians), the length of the arc subtended by that angle is almost equal to the length of the opposite side (sin θ) in the right triangle formed. This relationship is formally proven using the Squeeze Theorem by comparing the areas of a sector, a small triangle, and a larger triangle within the unit circle. This limit is the cornerstone that connects the geometry of circles to the rate of change in trigonometric functions.
5. Why must angles be in radians when proving and using trigonometric derivatives?
Using radians is essential because the fundamental limit proofs, such as lim(h→0) sin(h)/h = 1, are only valid when the angle 'h' is measured in radians. If angles were measured in degrees, the limit would become π/180. This would introduce a constant factor of (π/180) into all the derivative formulas, making them unnecessarily complex. For example, the derivative of sin(x°) would be (π/180)cos(x°). Radians keep the formulas simple and elegant, like d/dx(sin x) = cos x.
6. What is the difference in finding the derivative of sin(x²) versus sin²(x)?
The key difference lies in how the chain rule is applied. It is a common point of confusion for students.
- For sin(x²), the outer function is sin(u) and the inner function is u = x². The derivative is cos(x²) multiplied by the derivative of the inner function (2x). So, d/dx sin(x²) = 2x * cos(x²).
- For sin²(x), which is (sin x)², the outer function is u² and the inner function is u = sin x. The derivative is 2u multiplied by the derivative of the inner function (cos x). So, d/dx sin²(x) = 2(sin x) * cos x, which also equals sin(2x).
Correctly identifying the inner and outer functions is crucial to avoid errors.
7. How are proofs of trigonometric derivatives relevant to the CBSE Class 12 syllabus?
As per the CBSE Class 12 Maths syllabus for 2025-26, understanding the derivation of these formulas is a key learning objective within the 'Continuity and Differentiability' unit. The syllabus expects students to know the proof of d/dx(sin x) and d/dx(cos x) from first principles. The derivatives of tan(x), cot(x), sec(x), and csc(x) are expected to be derived using the quotient rule. Mastering these proofs is crucial for building a strong conceptual foundation for solving complex differentiation problems.
8. What is a real-world application of the derivative of a trigonometric function?
Derivatives of trigonometric functions are essential for describing any phenomenon involving oscillation or periodic motion. For example:
- In Physics, if the position of a simple pendulum is modelled by a cosine function, x(t) = A cos(ωt), its velocity is the derivative, v(t) = -Aω sin(ωt). This tells us the pendulum's instantaneous speed and direction.
- In Electrical Engineering, the voltage in an AC circuit is often a sine wave, V(t) = V₀ sin(ωt). Its derivative, the rate of change of voltage, is critical for analysing current and power.
These derivatives provide the mathematical language to analyse rates of change in any cyclical system.
