Exercise 10.1 | First Year Math 2025 | Allied Angles in Trigonometry

 Exercise 10.1 | First Year Math 2025 | Allied Angles in Trigonometry

Trigonometry is one of the most important branches of mathematics, especially for students preparing for higher studies in engineering, physics, and computer science. In the New First Year Math 2025, Exercise 10.1 introduces you to the concept of Allied Angles — an essential topic that helps in simplifying trigonometric expressions, solving equations, and understanding angle transformations.

This exercise focuses on the basic concept, definitions, and formulas of allied angles. We will explain each step in detail, give examples, and show practical uses so that you can master this topic for both exams and future applications.

1. What are Allied Angles?

In simple words, allied angles are angles that differ from each other by a multiple of 90° (or π/2 radians).

Mathematically:

If two angles differ by 90°, 180°, 270°, or 360° (or π/2, π, 3π/2, 2π in radians), they are called allied angles.

Example:

120° and 60° are allied angles because 120° = 180° - 60°

210° and 30° are allied angles because 210° = 180° + 30°

270° and 90° are allied angles because 270° = 360° - 90°

2. Why Learn Allied Angles?

Allied angles are important because they help us:

Simplify trigonometric functions for angles larger than 90°

Use standard values of trigonometric ratios for acute angles (0° to 90°) to find values for obtuse, reflex, or negative angles

Solve trigonometric equations

Understand periodicity and symmetry in trigonometric functions

In exams, allied angles save you time and reduce mistakes.

3. Quadrants in Trigonometry

Before learning formulas, you must understand the quadrant system. The coordinate plane is divided into 4 quadrants:

1. 1st Quadrant (0° to 90°) → All trigonometric functions are positive

2. 2nd Quadrant (90° to 180°) → sin and csc are positive; others are negative

3. 3rd Quadrant (180° to 270°) → tan and cot are positive; others are negative

4. 4th Quadrant (270° to 360°) → cos and sec are positive; others are negative

This rule is remembered as "All Students Take Calculus":

A → All positive (1st quadrant)

S → Sine positive (2nd quadrant)

T → Tangent positive (3rd quadrant)

C → Cosine positive (4th quadrant)

4. Basic Allied Angle Formulas

Let θ be an acute angle (0° ≤ θ ≤ 90°).

(a) Angles of form (90° ± θ):

sin(90° + θ) = cos θ

sin(90° - θ) = cos θ

cos(90° + θ) = -sin θ

cos(90° - θ) = sin θ

tan(90° + θ) = -cot θ

tan(90° - θ) = cot θ

(b) Angles of form (180° ± θ):

sin(180° + θ) = -sin θ

sin(180° - θ) = sin θ

cos(180° + θ) = -cos θ

cos(180° - θ) = -cos θ

tan(180° + θ) = tan θ

tan(180° - θ) = -tan θ

(c) Angles of form (270° ± θ):

sin(270° + θ) = -cos θ

sin(270° - θ) = -cos θ

cos(270° + θ) = sin θ

cos(270° - θ) = -sin θ

tan(270° + θ) = -cot θ

tan(270° - θ) = cot θ

(d) Angles of form (360° ± θ):

sin(360° + θ) = sin θ

sin(360° - θ) = -sin θ

cos(360° + θ) = cos θ

cos(360° - θ) = cos θ

tan(360° + θ) = tan θ

tan(360° - θ) = -tan θ

PDF SOLUTION :

5. How to Use Allied Angle Formulas

Step 1: Identify the given angle.

Step 2: Write it as one of the forms (90° ± θ, 180° ± θ, 270° ± θ, 360° ± θ).

Step 3: Determine the reference angle θ.

Step 4: Apply the allied angle formula.

Step 5: Decide the sign based on the quadrant.

6. Examples

Example 1: Find sin 120°

120° = 180° - 60° → use formula sin(180° - θ) = sin θ

sin 120° = sin 60° = √3/2

Example 2: Find cos 150°

150° = 180° - 30° → cos(180° - θ) = -cos θ

cos 150° = -cos 30° = -√3/2

Example 3: Find tan 210°

210° = 180° + 30° → tan(180° + θ) = tan θ

tan 210° = tan 30° = 1/√3

Example 4: Find sin 330°

330° = 360° - 30° → sin(360° - θ) = -sin θ

sin 330° = -sin 30° = -1/2

7. Allied Angles in Radians

In radians, the same concept applies.

90° = π/2

180° = π

270° = 3π/2

360° = 2π

Example: cos(π - θ) = -cos θ, sin(3π/2 + θ) = -cos θ

8. Common Mistakes to Avoid

Forgetting to check the sign based on the quadrant.

Mixing degrees and radians in one problem.

Using wrong reference angles.

Not simplifying the final trigonometric ratio.

9. Real-Life Applications of Allied Angles

Physics: Wave motion, alternating current (AC) circuits, and oscillations.

Engineering: Analysis of rotating machinery, robotics motion paths.

Astronomy: Calculating star positions and satellite orbits.

Computer Graphics: Rotation transformations and animation paths.

10. Why Allied Angles are Easy if Learned Systematically

If you remember:

1. Reference angle

2. Formula for transformation

3. Sign based on quadrant

…you can solve any allied angle question without memorizing dozens of values.

11. Summary Table

Angle Form Sine Cosine Tangent

90° - θ cos θ sin θ cot θ

90° + θ cos θ -sin θ -cot θ

180° - θ sin θ -cos θ -tan θ

180° + θ -sin θ -cos θ tan θ

270° - θ -cos θ -sin θ cot θ

270° + θ -cos θ sin θ -cot θ

360° - θ -sin θ cos θ -tan θ

360° + θ sin θ cos θ tan θ

12. Final Words

Exercise 10.1 in First Year Math 2025 gives you the foundation for all trigonometric transformations. Allied angles make solving trigonometric problems much faster and help you understand symmetry in the trigonometric circle. By practicing these formulas and applying quadrant rules, you will be able to solve even complex trigonometric equations with confidence.