Exercise 11.3 | Trigonometric Functions and Their Graphs | New First Year Math 2025 | Sir Khawar Mehmood | Math Universe Online

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In this detailed lesson, we will study Exercise 11.3 from the chapter Trigonometric Functions and Their Graphs.

This exercise focuses on two very important ideas in trigonometry —

👉 Maximum and Minimum values of trigonometric functions, and

👉 Real-world applications of trigonometric graphs.

This topic helps students understand how trigonometric functions like sin θ, cos θ, and tan θ behave in different situations, how they reach their highest and lowest values, and how these ideas can be used to solve practical mathematical problems.

🌟 Chapter Overview: Trigonometric Functions and Their Graphs

Trigonometric functions are among the most important functions in mathematics. They describe the relationship between the angles and sides of a triangle and have wide applications in physics, engineering, computer science, and even in daily life (like sound waves, light waves, and circular motion).

In this chapter, you learn how to:

Sketch and understand the graphs of trigonometric functions.

Determine the amplitude, period, and phase shift of different trigonometric functions.

Find the maximum and minimum values of given trigonometric expressions.

Apply these concepts to real-life situations.

🧮 Understanding Maximum and Minimum Values

Every trigonometric function has specific ranges — that means they can only take certain values.

1. Basic Trigonometric Ranges:

For sin θ,

→ Minimum value = −1

→ Maximum value = +1

→ Range = [−1, +1]

For cos θ,

→ Minimum value = −1

→ Maximum value = +1

→ Range = [−1, +1]

For tan θ,

→ It can take any real value (because tan θ = sin θ / cos θ and cos θ can be 0).

→ So, Range = (−∞, +∞)

These are the fundamental properties that help us find the maximum and minimum of more complicated trigonometric functions.

📘 Example 1: Simple Function

Let’s take an example:

f(θ) = 3sinθ + 2

Here, we know that:

−1 ≤ sinθ ≤ 1

Multiply the entire inequality by 3:

−3 ≤ 3sinθ ≤ 3

Now, add 2 on each side:

−3 + 2 ≤ 3sinθ + 2 ≤ 3 + 2

⇒ −1 ≤ f(θ) ≤ 5

✅ Hence,

Minimum value = −1

Maximum value = 5

This means the function f(θ) = 3sinθ + 2 oscillates between −1 and 5.

📗 Example 2: Function with Cosine

If we take f(θ) = 5cosθ − 4

Since −1 ≤ cosθ ≤ 1

Multiply by 5:

−5 ≤ 5cosθ ≤ 5

Subtract 4:

−5 − 4 ≤ 5cosθ − 4 ≤ 5 − 4

⇒ −9 ≤ f(θ) ≤ 1

✅ So,

Minimum value = −9

Maximum value = 1

📙 Example 3: Combination of Sine and Cosine

f(θ) = a sinθ + b cosθ

This type of function is very common in this exercise.

To find its maximum and minimum values, we use a special formula.

Formula:

\text{Maximum value} = \sqrt{a^2 + b^2}

\text{Minimum value} = -\sqrt{a^2 + b^2} 

🧩 Example:

f(θ) = 3sinθ + 4cosθ

⇒ √(3² + 4²) = √(9 + 16) = √25 = 5

✅ Maximum = +5, Minimum = −5

So this function oscillates between −5 and +5.

PDF SOLUTION:

  

📊 Graphical Representation

To understand these results better, you can plot the graphs of sinθ, cosθ, and their combinations.

Each trigonometric graph has a wave-like shape, repeating after a certain interval (called the period).

For sinθ and cosθ, the period = 2π

For tanθ, the period = π

The graph of y = sinθ starts from (0,0), goes up to +1, comes down to 0, then to −1, and returns again to 0.

The graph of y = cosθ starts from +1, drops to 0, goes to −1, and then returns to +1.

When we multiply or add constants, the graph shifts upward or downward, and its shape changes slightly — but the wave pattern always remains.

🧠 Real-World Applications of Trigonometric Graphs

Trigonometric graphs are not just theoretical concepts — they appear everywhere in science and technology.

1️⃣ Sound Waves

Sound travels in the form of waves. The shape of these waves is sinusoidal (same as the sine or cosine graph).

The amplitude shows the loudness of the sound, and the frequency shows how fast the sound vibrates.

2️⃣ Light and Heat Waves

The intensity of light or heat over time often follows a trigonometric pattern.

Physicists use trigonometric graphs to study reflection, refraction, and alternating current (AC).

3️⃣ Tides of the Sea

The height of tides changes in a wave-like pattern during the day.

Mathematically, the rise and fall of tides can be represented by a sine curve.

Example:

h(t) = 2sin(πt / 6) + 5

Here, h(t) shows the height of water (in meters) at time t (in hours).

4️⃣ Engineering and Mechanics

When a wheel rotates or a piston moves, its position often follows a trigonometric function.

In simple harmonic motion (SHM), displacement, velocity, and acceleration are all related to sine or cosine functions.

5️⃣ Electronics and Alternating Current (AC)

The voltage in an AC circuit changes with time in a sine wave form.

For example,

V(t) = V₀ sin(ωt)

where V₀ = peak voltage, ω = angular frequency, t = time.

This shows how math helps in designing electrical systems and circuits.

🧩 Amplitude, Period, and Phase Shift

To completely describe a trigonometric function, three things are very important:

1. Amplitude (A):

The height of the wave from the centerline.

For y = A sinθ or y = A cosθ,

Amplitude = |A|

2. Period (T):

The distance after which the graph repeats.

For y = sinθ or y = cosθ,

Period = 2π

For y = sin(kθ),

Period = (2π)/k

3. Phase Shift:

It tells how much the graph is shifted horizontally.

If y = sin(θ + α), the graph is shifted left by α.

If y = sin(θ − α), the graph is shifted right by α.

These parameters help you match real-world oscillations with exact trigonometric models.

🧾 Important Tips for Exercise 11.3

✅ Always begin by writing the range of the basic trigonometric function.

✅ Multiply or add constants carefully using inequality rules.

✅ For functions like a sinθ + b cosθ, use the formula √(a² + b²).

✅ Remember: sin²θ + cos²θ = 1 — this identity often helps in simplification.

✅ Practice graphing by hand — it helps you visualize the nature of the function.

✅ For application questions, identify the amplitude, period, and vertical shift from the given data.

🧾 Summary of Key Formulas

Function Maximum Minimum

sinθ +1 −1

cosθ +1 −1

a sinθ + b cosθ +√(a² + b²) −√(a² + b²)

k sinθ + c c + k c − k

k cosθ + c c + k c − k

💡 Why This Exercise Is Important

Exercise 11.3 builds the foundation for:

Calculus and Differentiation (finding slopes of curves),

Physics (wave motion, alternating current),

Engineering (rotational motion, vibration analysis), and

Mathematical Modelling (predicting periodic behavior).

When you understand how trigonometric functions behave, you can describe and predict many real-world systems — from the swinging of a pendulum to the motion of planets.

🎓 Learn with Sir Khawar Mehmood

In this lecture, Sir Khawar Mehmood explains every concept with clarity, step-by-step examples, and real-life applications.

You’ll learn:

How to find maximum and minimum values easily

How to transform trigonometric functions

How to apply graphs to daily life problems

Each question from Exercise 11.3 (New 2025 Edition) is solved in detail — ideal for Punjab Boards and First Year Math students.