Introduction to Trigonometry

12/13

Step 1: Identify sides - PQ = 12 cm (adjacent to P), QR = 5 cm (opposite to P). Step 2: Find hypotenuse PR using Pythagoras - PR = √(PQ² + QR²) = √(12² + 5²) = √(144 + 25) = √169 = 13 cm. Step 3: Apply cosine definition - cos P = adjacent/hypotenuse = PQ/PR. Step 4: Calculate cos P = 12/13. This is a 5-12-13 Pythagorean triple.

1/2

Step 1: Use the standard 30-60-90 triangle with sides in ratio 1:√3:2. Step 2: In this triangle, if hypotenuse = 2, then the side adjacent to 60° = 1. Step 3: cos 60° = adjacent/hypotenuse = 1/2. Step 4: This is a fundamental value to memorize. Note: cos 60° = sin 30° (complementary angles).

4/3

Step 1: Recall that cotangent is the reciprocal of tangent. Step 2: cot A = 1/tan A. Step 3: Given tan A = 3/4, so cot A = 1/(3/4). Step 4: Calculate cot A = 4/3. Remember: cot A and tan A are reciprocals, so cot A × tan A = 1.

1/2

Step 1: Recall standard values - sin 45° = 1/√2 and cos 45° = 1/√2. Step 2: Multiply these values - sin 45° × cos 45° = (1/√2) × (1/√2). Step 3: Calculate (1/√2) × (1/√2) = 1/2. Step 4: This product equals 1/2. Note: For 45° angle, sine and cosine are equal.