Relations Between Sides and Angles of a Triangle

30°

Step 1: Apply sine rule: a/sin A = b/sin B. Step 2: √3/sin 60° = 1/sin B. Step 3: sin 60° = √3/2, so √3/(√3/2) = 2 = 1/sin B. Step 4: sin B = 1/2. Step 5: Therefore B = 30°. Students often mistakenly choose 90° because sin 90° = 1, but 1/2 corresponds to 30°, not 90°.

120°

Step 1: The greatest angle is opposite the largest side c = 7. Step 2: Use cos C = (a² + b² - c²)/(2ab) with a = 3, b = 5, c = 7. Step 3: cos C = (9 + 25 - 49)/(2 × 3 × 5) = -15/30 = -1/2. Step 4: cos C = -1/2 means C = 120°. Step 5: The negative cosine confirms the angle is obtuse. A common error is taking a = 5, b = 7 instead of choosing correctly.

11/16

Step 1: Use cos B = (c² + a² - b²)/(2ac). Step 2: Substitute: cos B = (16 + 4 - 9)/(2 × 4 × 2). Step 3: Numerator = 11, Denominator = 16. Step 4: cos B = 11/16. Step 5: Note that 7/8 = 21/24 is actually cos A for this triangle; 11/16 is specifically for angle B. Always match the formula to the correct angle.

b = k sin B

Step 1: The sine rule states a/sin A = b/sin B = c/sin C = k. Step 2: From b/sin B = k, we multiply both sides by sin B. Step 3: This gives b = k sin B. Step 4: Similarly a = k sin A and c = k sin C. Step 5: This representation is very useful for proving identities — substituting a, b, c in terms of k and angles simplifies calculations greatly.