Start free
Back to the topic
Free printable worksheetlessoons.com

Mathematics, four units · Ages 16–17

Choosing Between the Sine Rule and the Cosine Rule

Name: Date:

For 11th grade math students. Decide which rule applies based on the given information, then calculate the missing value. Round answers to one decimal place.

  1. In triangle ABC, a = 12, angle A = 35 degrees, and angle B = 70 degrees. Identify which rule to use to find side b, and calculate it.

  2. In triangle ABC, b = 6, c = 10, and angle A = 45 degrees. Identify the rule to find side a, and calculate it.

  3. In triangle ABC, a = 8, b = 11, and c = 13. State which rule finds angle C, and find angle C.

  4. Calculate the area of a triangle with sides measuring 7 cm and 10 cm and an included angle of 40 degrees.

  5. In triangle ABC, angle A = 30 degrees, side a = 7, and side b = 12. Find the two possible values for angle B.

More topics and worksheets at lessoons.com© 2026 Lessons · 1 / 2
Answer keylessoons.com

Choosing Between the Sine Rule and the Cosine Rule

Answer key

  1. In triangle ABC, a = 12, angle A = 35 degrees, and angle B = 70 degrees. Identify which rule to use to find side b, and calculate it.

    Sine rule (matching pair A and a). b = (12 * sin(70)) / sin(35) ≈ 19.7.

  2. In triangle ABC, b = 6, c = 10, and angle A = 45 degrees. Identify the rule to find side a, and calculate it.

    Cosine rule (two sides and included angle). a^2 = 6^2 + 10^2 - 2(6)(10)cos(45) = 36 + 100 - 120(0.7071) ≈ 51.15. a ≈ 7.2.

  3. In triangle ABC, a = 8, b = 11, and c = 13. State which rule finds angle C, and find angle C.

    Cosine rule (all three sides known). cos(C) = (8^2 + 11^2 - 13^2) / (2 * 8 * 11) = (64 + 121 - 169) / 176 = 16 / 176 ≈ 0.0909. Angle C ≈ 84.8 degrees.

  4. Calculate the area of a triangle with sides measuring 7 cm and 10 cm and an included angle of 40 degrees.

    Area = 0.5 * 7 * 10 * sin(40) = 35 * 0.6428 ≈ 22.5 square cm.

  5. In triangle ABC, angle A = 30 degrees, side a = 7, and side b = 12. Find the two possible values for angle B.

    Sine rule: sin(B) / 12 = sin(30) / 7. sin(B) = (12 * 0.5) / 7 ≈ 0.8571. Acute B ≈ 59.0 degrees; obtuse B = 180 - 59.0 = 121.0 degrees. Both are valid because 30 + 121 < 180.