1st Year Math Chapter 10 Exercise 10.2 Notes: Trigonometric Functions of Any Angle, Fundamental Identities, and Signs (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 10.2 extends trigonometric functions beyond acute angles to any angle, using coordinates in the standard position, and introduces the fundamental identities and the rule for determining the sign of each function in every quadrant.
What Does This Exercise Cover?
This exercise teaches how to define sine, cosine, and the other trigonometric functions for any angle, the three core identities that connect them, and how to quickly determine whether a function’s value is positive or negative based on the angle’s quadrant.
Key Concepts
1. Coterminal Angles
Coterminal angles share the same terminal side when drawn in standard position, differing from each other by a multiple of 360° (or 2π radians).
2. Angle in Standard Position
An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis.
3. Trigonometric Functions of Any Angle
For a point (x, y) on the terminal side of an angle θ, at distance r from the origin, the trigonometric functions are defined as sin θ = y/r, cos θ = x/r, tan θ = y/x (and their reciprocals), regardless of the size of θ.
4. The Fundamental Identities
Three key identities hold for every angle θ: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ.
5. Signs of the Trigonometric Functions (the ASTC Rule)
The sign of each trigonometric function depends on the quadrant of the angle, following the ASTC pattern: All functions positive in Quadrant I, Sine (and cosecant) positive in Quadrant II, Tangent (and cotangent) positive in Quadrant III, Cosine (and secant) positive in Quadrant IV.
The ASTC Rule
| Quadrant | Positive Functions |
| I | All (sin, cos, tan, and reciprocals) |
| II | Sine and cosecant |
| III | Tangent and cotangent |
| IV | Cosine and secant |
Solved Examples
Example 1: Find a coterminal angle of 30° between 360° and 720°.
Add 360° to 30°: 30° + 360°.
Answer: 390°
Example 2: If sin θ = 3/5 and θ is in Quadrant II, find cos θ.
Using sin²θ + cos²θ = 1: cos²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5.
Since θ is in Quadrant II, cosine is negative there.
Answer: cos θ = −4/5
Example 3: Determine the sign of tan 200°.
200° lies between 180° and 270°, placing it in Quadrant III.
By the ASTC rule, tangent is positive in Quadrant III.
Answer: tan 200° is positive
Sample MCQs
1. Two angles are coterminal if they:
a) Have exactly the same value b) Differ by a multiple of 360° c) Are complementary d) Are supplementary
Answer: b) Differ by a multiple of 360°
2. Which of these is a correct fundamental identity?
a) sin²θ − cos²θ = 1 b) sin²θ + cos²θ = 1 c) sin θ + cos θ = 1 d) sin θ × cos θ = 1
Answer: b) sin²θ + cos²θ = 1
3. In which quadrant are all trigonometric functions positive?
a) I b) II c) III d) IV
Answer: a) I
4. In Quadrant II, which trigonometric function (and its reciprocal) is positive?
a) Sine (and cosecant) b) Cosine (and secant) c) Tangent (and cotangent) d) None
Answer: a) Sine (and cosecant)
5. The identity 1 + tan²θ is equal to:
a) sin²θ b) cos²θ c) sec²θ d) csc²θ
Answer: c) sec²θ
Important Short Questions
- Define coterminal angles.
- State the three fundamental trigonometric identities.
- State the ASTC rule for the signs of trigonometric functions.
- Find a coterminal angle of 50°, other than 50° itself, between −360° and 360°.
- Determine the sign of cos 300°.
Important Long Questions
- If cos θ = −5/13 and θ is in Quadrant III, find sin θ and tan θ using the fundamental identities.
- Determine the quadrant and the sign of all six trigonometric functions for an angle of 250°.
- Prove that 1 + cot²θ = csc²θ, starting from the identity sin²θ + cos²θ = 1.
- Explain how the ASTC rule follows from the signs of the x and y coordinates in each quadrant.
How to Approach This Exercise Effectively
- Draw a quick sketch of the angle’s terminal side whenever you’re unsure of its quadrant — visualizing (x, y) signs makes ASTC automatic.
- Memorize the ASTC rule with a simple mnemonic (like ‘All Students Take Calculus’) to recall it instantly during exams.
- When given one trigonometric value and asked for another, identify the quadrant first — this determines the correct sign before you even calculate the magnitude.
- Practice deriving 1+tan²θ=sec²θ and 1+cot²θ=csc²θ from sin²θ+cos²θ=1 by dividing through by cos²θ or sin²θ respectively — this builds real understanding, not just memorization.
- Keep the definitions sin θ = y/r, cos θ = x/r, tan θ = y/x in mind as the ultimate source of truth whenever a sign or value seems unclear.
FAQs
Q: What does it mean for two angles to be coterminal?
A: It means they end up pointing in exactly the same direction when drawn from the same starting position, even though their angle measures differ by a full rotation (or rotations).
Q: Why are the fundamental identities always true, regardless of the angle?
A: They follow directly from the Pythagorean relationship between x, y, and r (x² + y² = r²) that defines the coordinates of any point on a circle of radius r.
Q: How can the ASTC rule be remembered easily?
A: A common memory aid is the phrase ‘All Students Take Calculus,’ where each word’s first letter matches the positive function set in quadrants I through IV in order.
Q: Can trigonometric functions be defined for angles greater than 360°?
A: Yes — since angles greater than 360° are simply coterminal with an angle between 0° and 360°, their trigonometric values are identical to that coterminal angle’s values.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 10: Trigonometric Identities, Exercise 10.2.