Trigonometry — Trigonometric Ratios and Identities

Overview

Trigonometry is a fundamental branch of mathematics that deals with the relationships between angles and sides of triangles, particularly right-angled triangles. For the JKTET Paper II, this topic forms a crucial component of the mathematics section and tests your understanding of basic trigonometric ratios, their relationships, and standard identities.

Mastery of trigonometry is essential because it connects geometry with algebra and has practical applications in measurement, surveying, and physics problems. Questions typically test your ability to calculate ratio values for standard angles, apply identities to simplify expressions, and prove trigonometric relationships. Students who thoroughly understand the six ratios and the three fundamental identities can handle most exam questions with confidence.

The scope for JKTET is limited to right-triangle trigonometry and standard identities — you are not expected to deal with trigonometric equations, graphs, or inverse functions at this level.

Key Concepts

• **Right-angled triangle reference**: All six trigonometric ratios are defined with respect to an acute angle (θ) in a right-angled triangle, using three sides — opposite, adjacent, and hypotenuse.

• **Six trigonometric ratios**: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). The last three are reciprocals of the first three.

• **Complementary angle relationship**: The trigonometric ratio of an angle equals the co-ratio of its complement. For example, sin(90° − θ) = cos θ.

• **Standard angles**: You must memorise exact values for 0°, 30°, 45°, 60°, and 90°. These appear repeatedly in calculations.

• **Pythagorean identities**: Three fundamental identities derived from the Pythagorean theorem connect the squares of ratios.

• **Ratio relationships**: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. These help convert between ratios.

• **Range restrictions**: sin θ and cos θ always lie between −1 and 1. For acute angles in a right triangle, all six ratios are positive.

Formulas / Key Facts

### Definitions of Six Ratios (for acute angle θ in right triangle)

| Ratio | Formula | |-------|---------| | sin θ | Opposite / Hypotenuse | | cos θ | Adjacent / Hypotenuse | | tan θ | Opposite / Adjacent | | cosec θ | Hypotenuse / Opposite = 1 / sin θ | | sec θ | Hypotenuse / Adjacent = 1 / cos θ | | cot θ | Adjacent / Opposite = 1 / tan θ |

### Standard Angle Values

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

**Memory trick**: For sin values of 0°, 30°, 45°, 60°, 90°, think √0/2, √1/2, √2/2, √3/2, √4/2.

### Three Pythagorean Identities

1. sin²θ + cos²θ = 1 2. 1 + tan²θ = sec²θ 3. 1 + cot²θ = cosec²θ

### Complementary Angle Relationships

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

### Quotient Identities

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

Worked Examples

### Example 1: Finding ratio values from a triangle

**Problem**: In a right triangle, the side opposite to angle A is 5 cm and the hypotenuse is 13 cm. Find sin A, cos A, and tan A.

**Solution**:

• Opposite = 5, Hypotenuse = 13
• Adjacent = √(13² − 5²) = √(169 − 25) = √144 = 12
• sin A = 5/13
• cos A = 12/13
• tan A = 5/12

### Example 2: Using standard angle values

**Problem**: Evaluate: sin²60° + cos²60° + tan²45°

**Solution**:

• sin 60° = √3/2, so sin²60° = 3/4
• cos 60° = 1/2, so cos²60° = 1/4
• tan 45° = 1, so tan²45° = 1
• Sum = 3/4 + 1/4 + 1 = 1 + 1 = **2**

### Example 3: Proving an identity

**Problem**: Prove that (1 − sin²θ) × sec²θ = 1

**Solution**:

• From identity: 1 − sin²θ = cos²θ
• So LHS = cos²θ × sec²θ
• Since sec θ = 1/cos θ, we have sec²θ = 1/cos²θ
• LHS = cos²θ × (1/cos²θ) = 1 = RHS

### Example 4: Complementary angles

**Problem**: If sin 5A = cos(A − 6°), find A.

**Solution**:

• Using sin θ = cos(90° − θ), we can write:
• sin 5A = cos(90° − 5A)
• So cos(90° − 5A) = cos(A − 6°)
• Therefore: 90° − 5A = A − 6°
• 90° + 6° = A + 5A
• 96° = 6A
• **A = 16°**

Common Mistakes

• **Confusing opposite and adjacent**: Students often mix up which side is opposite and which is adjacent. The opposite side faces the angle; the adjacent side touches the angle (but is not the hypotenuse). → Always mark the angle first, then identify sides relative to it.

• **Using wrong standard values**: Mixing up sin 30° with sin 60° is extremely common. → Remember that sin increases from 0° to 90° (0, 1/2, 1/√2, √3/2, 1), while cos decreases over the same range.

• **Forgetting to square properly**: Writing sin²60° = √3/2 instead of (√3/2)² = 3/4. → When you see sin²θ, first find sin θ, then square the entire value.

• **Dividing by zero in tan 90° or cot 0°**: These are undefined, not zero or infinity in exam answers. → Flag these as "not defined" when they appear.

• **Misapplying identities**: Using sin²θ + cos²θ = 1 but forgetting the corresponding forms with tan-sec or cot-cosec. → Practice deriving all three from the basic identity by dividing by sin²θ or cos²θ.

Quick Reference

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

• sin²θ + cos²θ = 1 — the master identity from which others derive

• sin 30° = cos 60° = 1/2; sin 60° = cos 30° = √3/2; sin 45° = cos 45° = 1/√2

• tan θ × cot θ = 1; sin θ × cosec θ = 1; cos θ × sec θ = 1

• For complementary angles: "co-" ratios swap (sin ↔ cos, tan ↔ cot, sec ↔ cosec)

• tan²θ + 1 = sec²θ — useful when simplifying expressions involving tan and sec