What is Trigonometry in Optimisation? | Simple Explanation for Class 10

S6-SA2-0461

What is the Use of Trigonometry in Optimisation Problems (calculus based, basic)?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Trigonometry helps us find the best possible angles or dimensions in problems where we want to maximise or minimise something, like area or distance. When combined with calculus, it allows us to precisely locate these optimal points by using derivatives of trigonometric functions.

Simple Example

Quick Example

Imagine you want to build a ramp for a toy car that reaches the highest point possible with a fixed length of wood. Trigonometry helps define the ramp's angle and height. Optimisation using calculus then helps you find the exact angle that gives the maximum height.

Worked Example

Step-by-Step

PROBLEM: A farmer wants to fence a rectangular plot next to a river. He has 100 metres of fencing and doesn't need to fence the side along the river. What dimensions will give the maximum area?

Step 1: Define variables. Let the two sides perpendicular to the river be 'x' and the side parallel to the river be 'y'.
---Step 2: Formulate the perimeter and area equations. Perimeter: 2x + y = 100. Area: A = x * y.
---Step 3: Express Area in terms of one variable. From perimeter, y = 100 - 2x. Substitute this into the area equation: A(x) = x * (100 - 2x) = 100x - 2x^2.
---Step 4: Use calculus to find the maximum. To maximise A(x), find its derivative with respect to x: A'(x) = d/dx (100x - 2x^2) = 100 - 4x.
---Step 5: Set the derivative to zero and solve for x. 100 - 4x = 0 => 4x = 100 => x = 25 metres.
---Step 6: Find y using the perimeter equation. y = 100 - 2(25) = 100 - 50 = 50 metres.
---Step 7: Calculate the maximum area. A = 25 * 50 = 1250 square metres.
ANSWER: The dimensions for maximum area are 25 metres by 50 metres, giving an area of 1250 square metres.

Why It Matters

This concept is crucial in designing efficient machines, planning satellite orbits, and even creating realistic graphics in video games. Engineers use it to build stronger bridges, scientists use it to understand wave patterns, and even doctors use it in medical imaging for better diagnoses.

Common Mistakes

MISTAKE: Forgetting to express the quantity to be optimised (like area or volume) in terms of a single variable. | CORRECTION: Always use the constraint equation (like total fencing or material) to substitute and eliminate one variable from the main function you want to optimise.

MISTAKE: Not setting the derivative to zero to find critical points. | CORRECTION: The maximum or minimum of a function often occurs where its rate of change (derivative) is zero, so this step is essential.

MISTAKE: Confusing the maximum/minimum value of the function with the variable values that produce it. | CORRECTION: After finding the 'x' or 'theta' value, make sure to plug it back into the original function to find the actual maximum or minimum quantity (e.g., maximum area, not just the side length).

Practice Questions

Try It Yourself

QUESTION: A wire of length 20 cm is bent to form a rectangle. What is the maximum possible area of this rectangle? | ANSWER: 25 square cm

QUESTION: A cylindrical can needs to hold 100 cubic cm of juice. What dimensions (radius and height) will minimise the amount of material needed to make the can (i.e., minimise surface area)? Assume the can has a top and bottom. (Hint: Volume V = pi * r^2 * h, Surface Area A = 2 * pi * r^2 + 2 * pi * r * h) | ANSWER: Radius = cube_root(50/pi) cm, Height = 2 * cube_root(50/pi) cm

QUESTION: A person standing on the ground throws a ball. The height 'h' of the ball (in metres) after 't' seconds is given by h(t) = 20t - 5t^2. What is the maximum height the ball reaches? | ANSWER: 20 metres

MCQ

Quick Quiz

Which mathematical tool is primarily used to find the exact point of maximum or minimum in an optimisation problem?

Algebraic equations

Trigonometric identities

Derivatives (Calculus)

Geometric shapes

The Correct Answer Is:

Derivatives from calculus help us find the rate of change of a function, and setting the derivative to zero allows us to locate points where the function reaches its maximum or minimum value. Trigonometric identities might be part of the function, but derivatives are the tool for optimisation.

Real World Connection

In the Real World

From ISRO scientists optimising rocket launch trajectories to e-commerce companies like Flipkart or Amazon planning the most efficient delivery routes, optimisation with trigonometry and calculus is everywhere. Even in cricket, analysts use these principles to find the optimal angle for a shot to cover maximum distance or clear the boundary.

Key Vocabulary

Key Terms

OPTIMISATION: The process of finding the best solution (maximum or minimum value) for a problem | DERIVATIVE: A tool in calculus that measures how a function changes as its input changes; it tells us the slope of the tangent line | CRITICAL POINT: A point where the derivative of a function is zero or undefined, often indicating a maximum or minimum | TRIGONOMETRIC FUNCTION: Functions like sine, cosine, tangent that relate angles to side lengths of right-angled triangles

What's Next

What to Learn Next

Next, explore 'Applications of Derivatives' in more detail. You'll learn how to use the second derivative test to confirm if a critical point is a maximum or minimum, which is a crucial step in advanced optimisation problems. Keep practising!