Introduction to Trigonometry

Overview

Trigonometry forms a foundational bridge between geometry and algebra, dealing with relationships between angles and sides of triangles. For Bihar TET Paper II, this topic carries significant weight as it tests both conceptual understanding and computational accuracy. The syllabus focuses specifically on trigonometric ratios and basic identities—content that appears regularly in the mathematics section.

Students must master the six trigonometric ratios, understand their relationships in right-angled triangles, and apply fundamental identities to simplify expressions and solve problems. This topic connects directly to mensuration, geometry, and later topics like heights and distances. A clear grasp here builds confidence for more complex applications in teaching practice.

The key to success lies in memorising the ratio definitions, understanding the reciprocal relationships, and practising identity-based simplifications until they become automatic.

Key Concepts

• **Right-angled triangle orientation**: In any right triangle, identify the hypotenuse (longest side, opposite the 90° angle), the perpendicular (side opposite to the angle in question), and the base (side adjacent to the angle, excluding hypotenuse).

• **Six trigonometric ratios are paired**: Sine-Cosecant, Cosine-Secant, and Tangent-Cotangent form reciprocal pairs. Understanding one ratio in each pair automatically gives you the other.

• **Ratios depend on angle, not triangle size**: For a given angle θ, the trigonometric ratios remain constant regardless of how large or small the right triangle is—this is the fundamental principle that makes trigonometry useful.

• **Complementary angle relationship**: The sine of an angle equals the cosine of its complement, and vice versa. This means sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.

• **The Pythagorean identity is the master identity**: sin²θ + cos²θ = 1 is derived directly from the Pythagorean theorem and forms the basis for the other two fundamental identities.

• **Standard angles (0°, 30°, 45°, 60°, 90°)**: These five angles have exact ratio values that must be memorised—they appear in almost every trigonometry problem.

• **Domain restrictions**: Tan θ and sec θ are undefined at 90°; cot θ and cosec θ are undefined at 0°. This is because division by zero occurs in these cases.

Formulas / Key Facts

**Six Trigonometric Ratios** (for angle θ in a right triangle):

• sin θ = Perpendicular / Hypotenuse = P/H
• cos θ = Base / Hypotenuse = B/H
• tan θ = Perpendicular / Base = P/B = sin θ / cos θ

**Reciprocal Ratios**:

• cosec θ = 1 / sin θ = H/P
• sec θ = 1 / cos θ = H/B
• cot θ = 1 / tan θ = B/P = cos θ / sin θ

**Three Fundamental Identities**:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

**Standard Angle Values Table**:

| Angle | 0° | 30° | 45° | 60° | 90° | |-------|-----|------|------|------|------| | sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 | | cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 | | tan θ | 0 | 1/√3 | 1 | √3 | undefined |

**Memory trick for sin values**: Write 0, 1, 2, 3, 4 under the angles, then take square root and divide by 2: √0/2, √1/2, √2/2, √3/2, √4/2.

**Complementary Angle Relations**:

• sin(90° − θ) = cos θ
• tan(90° − θ) = cot θ
• sec(90° − θ) = cosec θ

Worked Examples

**Example 1**: If sin θ = 3/5, find the values of cos θ and tan θ.

*Solution*: Using sin²θ + cos²θ = 1 (3/5)² + cos²θ = 1 9/25 + cos²θ = 1 cos²θ = 1 − 9/25 = 16/25 cos θ = 4/5 (taking positive value for acute angle)

Now, tan θ = sin θ / cos θ = (3/5) / (4/5) = 3/4

*Alternative method*: Since sin θ = P/H = 3/5, we have P = 3, H = 5. By Pythagoras: B² = H² − P² = 25 − 9 = 16, so B = 4. Therefore, cos θ = B/H = 4/5 and tan θ = P/B = 3/4.

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**Example 2**: Prove that (1 + tan²θ) × cos²θ = 1

*Solution*: LHS = (1 + tan²θ) × cos²θ Using identity: 1 + tan²θ = sec²θ = sec²θ × cos²θ = (1/cos²θ) × cos²θ = 1 = RHS

Hence proved.

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**Example 3**: Evaluate: sin 60° × cos 30° + sin 30° × cos 60°

*Solution*: Substituting standard values: = (√3/2) × (√3/2) + (1/2) × (1/2) = 3/4 + 1/4 = 4/4 = 1

*Note*: This is actually sin(60° + 30°) = sin 90° = 1, demonstrating the sine addition pattern.

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**Example 4**: If tan θ = 12/5, find sec θ.

*Solution*: Using 1 + tan²θ = sec²θ 1 + (12/5)² = sec²θ 1 + 144/25 = sec²θ 25/25 + 144/25 = sec²θ 169/25 = sec²θ sec θ = 13/5

Common Mistakes

• **Confusing base and perpendicular**: Students often mix up which side is the base and which is the perpendicular. The perpendicular is always opposite to the angle θ, and the base is adjacent to θ (not the hypotenuse). → Always mark the angle first, then identify sides relative to that specific angle.

• **Forgetting that tan 90° is undefined**: Many students write tan 90° = 1 or some other value. → Remember that tan θ = sin θ / cos θ, and since cos 90° = 0, division by zero makes tan 90° undefined.

• **Applying identities with wrong signs**: Writing sin²θ − cos²θ = 1 instead of sin²θ + cos²θ = 1. → The Pythagorean theorem uses addition (a² + b² = c²), so the identity derived from it also uses addition.

• **Rationalising errors**: When simplifying expressions with √2 or √3 in denominators, students make calculation errors. → Always rationalise step-by-step: multiply numerator and denominator by the same surd.

• **Ignoring complementary relationships**: Not recognising that sin 60° = cos 30° leads to longer calculations. → Train yourself to spot complementary pairs—they simplify problems significantly.

Quick Reference

• **SOH-CAH-TOA**: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

• **Reciprocal pairs**: cosec = 1/sin, sec = 1/cos, cot = 1/tan.

• **Master identity**: sin²θ + cos²θ = 1 (all other identities derive from this).

• **At 45°**: sin = cos = 1/√2; tan = 1.

• **At 30° and 60°**: sin and cos swap; tan 30° = 1/√3, tan 60° = √3.

• **Quick check**: sin θ and cos θ always lie between 0 and 1 for acute angles.