Exercise 10.3 – Half Angle & Double Angle Formulas | First Year Mathematics 2025

 Exercise 10.3 – Half Angle & Double Angle Formulas | First Year Mathematics 2025

Introduction

Trigonometry is a major part of first-year mathematics (FSc Part 1 or equivalent) and plays a big role in board exam preparation. In the New First Year Math 2025 Edition, Exercise 10.3 focuses entirely on Half Angle and Double Angle Formulas.

These formulas allow you to:

Find the trigonometric values of an angle when you know double or half of it.

Simplify complex trigonometric expressions.

Prove important trigonometric identities.

Solve problems that appear regularly in board exams.

This exercise is very important for both short and long questions. Many MCQs are also made from this topic.

Double Angle Formulas

The double angle means 2θ. These formulas come directly from sum formulas:

1. sin 2θ = 2 sin θ cos θ

2. cos 2θ = cos²θ – sin²θ

Alternate forms:

cos 2θ = 1 – 2 sin²θ

cos 2θ = 2 cos²θ – 1

3. tan 2θ = (2 tan θ) / (1 – tan²θ)

Tip: You can choose the form of cos 2θ that makes your problem easier to solve.

Half Angle Formulas

The half angle means θ/2. These formulas are derived from double angle formulas:

1. sin(θ/2) = ± √((1 – cos θ) / 2)

2. cos(θ/2) = ± √((1 + cos θ) / 2)

3. tan(θ/2) = ± √((1 – cos θ) / (1 + cos θ))

Other useful forms for tan(θ/2):

tan(θ/2) = sin θ / (1 + cos θ)

tan(θ/2) = (1 – cos θ) / sin θ

Important: The ± (plus/minus) sign depends on which quadrant θ/2 lies in.

Where You Will Use These Formulas

In Exercise 10.3, you will:

Prove trigonometric identities using half angle and double angle formulas.

Change given expressions into simpler forms.

Solve equations where the angles are in the form of 2θ or θ/2.

Work with problems where the quadrant decides the sign of the answer.

Step-by-Step Method to Solve

1. Identify the given angle type – See if it’s 2θ or θ/2.

2. Select the right formula – Choose the version that will make the problem simpler.

3. Convert into sine and cosine – This often makes proofs easier.

4. Apply basic identities – Use sin²θ + cos²θ = 1 to simplify powers.

5. Simplify step-by-step – Never skip steps.

6. Check the quadrant – For half angle problems, decide whether to use positive or negative value.

Example 1 – Proving with Double Angle

Prove: sin 2θ = 2 sin θ cos θ

Solution:

We know: sin(A + B) = sin A cos B + cos A sin B

Put A = θ and B = θ:

sin(θ + θ) = sin θ cos θ + cos θ sin θ

sin 2θ = 2 sin θ cos θ

Hence proved.

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Example 2 – Proving cos 2θ Form

Prove: cos 2θ = 1 – 2 sin²θ

Solution:

We know: cos 2θ = cos²θ – sin²θ

Replace cos²θ with (1 – sin²θ):

cos 2θ = (1 – sin²θ) – sin²θ

cos 2θ = 1 – 2 sin²θ

Hence proved.

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Example 3 – Using Half Angle Formula

Find: sin(θ/2) when cos θ = 1/2 and 0 < θ < π

Solution:

Formula: sin(θ/2) = ± √((1 – cos θ) / 2)

Since θ = 60° (cos θ = 1/2) and θ/2 = 30°, sin is positive:

sin(θ/2) = √((1 – 1/2) / 2)

sin(θ/2) = √((1/2) / 2)

sin(θ/2) = √(1/4)

sin(θ/2) = 1/2

Example 4 – Using tan(θ/2) Alternative Form

Prove: tan(θ/2) = sin θ / (1 + cos θ)

Solution:

From double angle: sin θ = 2 sin(θ/2) cos(θ/2)

Also: cos θ = cos²(θ/2) – sin²(θ/2)

Now: sin θ / (1 + cos θ)

= [2 sin(θ/2) cos(θ/2)] / [1 + cos²(θ/2) – sin²(θ/2)]

Using cos² – sin² = cos 2A and 1 + cos² – sin² = 2 cos²(θ/2):

= [2 sin(θ/2) cos(θ/2)] / [2 cos²(θ/2)]

= tan(θ/2)

PDF SOLUTION:

Hence proved.

Why Exercise 10.3 Is Important for Exams

Long questions – Many proofs and simplifications in board exams come from here.

Short questions – Direct formula use is common in objective type.

MCQs – Quick application of formulas needed.

Past papers – Many questions are repeated with small changes.

Common Mistakes to Avoid

1. Wrong sign in half angle formulas – Always check the quadrant.

2. Mixing up formulas – Example: using sin²θ formula in place of cos²θ formula.

3. Skipping steps – This leads to calculation errors.

4. Not simplifying – Some students stop halfway without fully proving LHS = RHS.

Preparation Tips

Memorize all double and half angle formulas daily.

Practice writing each proof step-by-step without looking at the book.

Solve past paper questions for speed and accuracy.

Test yourself by solving random questions within a set time.

Conclusion

Exercise 10.3 in the New First Year Math 2025 Edition is a core trigonometry exercise based on half angle and double angle formulas. By mastering these formulas and their variations, you can easily handle both proof-based and simplification questions.

Consistent practice will not only help you score well in board exams but also prepare you for advanced mathematics in higher classes. Remember, the key is to know the formulas by heart, apply them correctly, and check the signs based on the quadrant.