1st Year Math Chapter 10 Exercise 10.3 Notes: Trigonometric Functions of Special and Quadrantal Angles (Punjab Board 2026-27)

Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.

Exercise 10.3 focuses on memorizing and applying the exact trigonometric values of the special acute angles (30°, 45°, 60°) and the quadrantal angles (0°, 90°, 180°, 270°, 360°), which appear constantly throughout trigonometry.

What Does This Exercise Cover?

This exercise builds fluency with the exact trigonometric values that come from special right triangles and from angles lying exactly on the coordinate axes, so they can be recalled instantly rather than recalculated each time.

Key Concepts

1. Special Acute Angles (30°, 45°, 60°)

The exact trigonometric values of 30°, 45°, and 60° come from two special right triangles: the 45°-45°-90° triangle (an isosceles right triangle) and the 30°-60°-90° triangle (half of an equilateral triangle).

2. Quadrantal Angles (0°, 90°, 180°, 270°, 360°)

Quadrantal angles lie exactly on one of the coordinate axes. Their trigonometric values are found directly from the coordinates of the point where the terminal side meets the axis, rather than from a triangle.

Trigonometric Values of Special Acute Angles

Trigonometric Values of Quadrantal Angles

Solved Examples

Example 1: Find the exact value of sin 60° + cos 30°.

sin 60° = √3/2, cos 30° = √3/2.

Sum: √3/2 + √3/2.

Answer: √3

Example 2: Evaluate tan 45° × sin 90°.

tan 45° = 1, sin 90° = 1.

Product: 1 × 1.

Answer: 1

Example 3: Find the value of cos 180° + sin 270°.

cos 180° = −1, sin 270° = −1.

Sum: (−1) + (−1).

Answer: −2

Sample MCQs

1. The value of sin 30° is:

a) 1/2   b) √3/2   c) 1   d) 0

Answer: a) 1/2

2. The value of cos 90° is:

a) 1   b) 0   c) −1   d) Undefined

Answer: b) 0

3. The value of tan 45° is:

a) 0   b) 1   c) Undefined   d) √3

Answer: b) 1

4. The value of sin 180° is:

a) 1   b) −1   c) 0   d) Undefined

Answer: c) 0

5. The value of cos 60° is:

a) 1/2   b) √3/2   c) 1   d) √2/2

Answer: a) 1/2

Important Short Questions

• State the exact values of sin 45°, cos 45°, and tan 45°.
• State the exact values of sin 30° and cos 60°. What do you notice about them?
• Evaluate cos 0° + sin 90°.
• What is the value of tan 90°? Explain why.
• Evaluate sin 270°.

Important Long Questions

• Find the exact value of sin 60° × cos 30° + cos 60° × sin 30°.
• Evaluate cos²45° + sin²45°, and confirm the result satisfies the fundamental identity from Exercise 10.2.
• Find the exact value of tan 60° − tan 30°, simplifying your answer fully.
• Explain why tan 90° is undefined, using the definition of tangent in terms of sine and cosine.

How to Approach This Exercise Effectively

1. Learn the 30-60-90 and 45-45-90 triangles’ side ratios by heart — every special angle value can be re-derived from them if memory fails under pressure.
2. Notice the symmetry between 30° and 60° (their sine and cosine values swap) — this halves the amount you need to memorize.
3. For quadrantal angles, picture the point where the terminal side crosses the axis and read off (x, y) directly, rather than memorizing values by rote.
4. Whenever a trigonometric ratio has a zero denominator (like tan 90° = sin90°/cos90° = 1/0), immediately identify it as undefined rather than trying to compute a number.
5. Practice combining special-angle values in short expressions (like sums or products) since these are common in short-question exams.

FAQs

Q: Why are 30°, 45°, and 60° called ‘special angles’?

A: Because their trigonometric values can be found exactly, using simple right-triangle geometry, rather than needing a calculator or approximation.

Q: How can the values of these special angles be derived geometrically?

A: By constructing a 45-45-90 triangle (from a square’s diagonal) or a 30-60-90 triangle (from an equilateral triangle’s height), then reading off side ratios to find sine, cosine, and tangent.

Q: Why are some trigonometric values undefined at certain angles?

A: Because tangent, cotangent, secant, and cosecant are defined as ratios, and those ratios have a zero denominator at specific angles, making the value undefined rather than a real number.

Q: Is there a pattern connecting sine and cosine values of complementary angles?

A: Yes — sin θ always equals cos(90° − θ), which is why sin 30° = cos 60° and sin 60° = cos 30°.

Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 10: Trigonometric Identities, Exercise 10.3.