Loading...

About Trigonometry Applications — Class 10 IB

Apply sine and cosine rules, solve non-right triangles, and work with trigonometric graphs. This topic is part of the IB Class 10 mathematics syllabus (chapter: Unit 4). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Trigonometry Applications — solved examples for Class 10 IB

Example 1easy

In a non-right-angled triangle PQR, if we know the lengths of sides p and q, and the measure of angle R (the angle included between sides p and q), which rule is most appropriate to find the length of side r?
  1. A)Cosine Rule
  2. B)Sine Rule
  3. C)Pythagorean Theorem
  4. D)Area Formula (1/2ab sin C)

Step-by-step solution

  1. The problem provides two sides (p and q) and the included angle (R). This is known as the Side-Angle-Side (SAS) case.
  2. The Cosine Rule is specifically designed to find the third side when two sides and the included angle are known, or to find an angle when all three sides are known.
  3. The formula for finding side r would be r² = p² + q² - 2pq cos R.

Answer: Cosine Rule

Example 2medium

In triangle ABC, angle A = 45°, angle B = 60°, and side a = 10 cm. Which expression correctly represents the length of side b?
  1. A)10 sin 60° / sin 45°
  2. B)10 sin 45° / sin 60°
  3. C)10 sin 60° × sin 45°
  4. D)10 / (sin 45° × sin 60°)

Step-by-step solution

  1. The Sine Rule states that for any triangle, a/sin A = b/sin B = c/sin C.
  2. Given a = 10 cm, A = 45°, and B = 60°, we want to find b.
  3. Using the Sine Rule: 10 / sin 45° = b / sin 60°.
  4. Rearranging to solve for b: b = (10 × sin 60°) / sin 45°.

Answer: 10 sin 60° / sin 45°

Example 3hard

In a triangle ABC, if the relationship a² = b² + c² - bc holds, where a, b, and c are the lengths of the sides opposite angles A, B, and C respectively, which of the following statements must be true?
  1. A)Angle A is 30°
  2. B)Angle A is 60°
  3. C)Angle A is 90°
  4. D)The triangle is isosceles

Step-by-step solution

  1. The Cosine Rule states that a² = b² + c² - 2bc cosA.
  2. Comparing the given relation a² = b² + c² - bc with the Cosine Rule, we can equate the terms involving 'bc': -bc = -2bc cosA.
  3. Dividing both sides by -bc (since b and c are side lengths, bc ≠ 0), we get 1 = 2cosA.
  4. Therefore, cosA = 1/2. The angle A for which cosA = 1/2 is 60° (within the context of a triangle, 0° < A < 180°).

Answer: Angle A is 60°

Practice questions on Trigonometry Applications

  1. Q1.easy

    Consider a triangle ABC where angle A = 45°, angle B = 60°, and side a (opposite angle A) = 10 cm. Which of the following statements about finding side b is correct?
    1. A)The Cosine Rule must be used because we have two angles.
    2. B)The Pythagorean Theorem can be used if we draw an altitude.
    3. C)The Sine Rule is the most direct method as we have an angle and its opposite side, plus another angle.
    4. D)Side b cannot be found with the given information.
    Show answer

    Answer: The Sine Rule is the most direct method as we have an angle and its opposite side, plus another angle.

    Hint: Recall the conditions under which the Sine Rule is most effectively applied. What information do you have?

  2. Q2.easy

    A student wants to calculate the area of a triangle XYZ. They know the lengths of sides x = 8 cm and y = 12 cm, and the measure of the included angle Z = 30°. Which expression correctly represents the area of triangle XYZ?
    1. A)Area = (1/2) × 8 × 12
    2. B)Area = (1/2) × 8 × 12 × sin(30°)
    3. C)Area = 8 × 12 × sin(30°)
    4. D)Area = (1/2) × 8 × 12 × cos(30°)
    Show answer

    Answer: Area = (1/2) × 8 × 12 × sin(30°)

    Hint: Remember the formula for the area of a non-right-angled triangle when two sides and the included angle are known.

  3. Q3.easy

    Ravi was solving for side 'c' in a triangle ABC where a = 5 cm, b = 7 cm, and angle C = 60°. He wrote the equation as: c² = 5² + 7² - 2 × 5 × 7 × sin(60°). What is the mistake in Ravi's approach?
    1. A)He should have used the Sine Rule instead of the Cosine Rule.
    2. B)The angle C should be opposite side c, not included between a and b.
    3. C)The formula for c² should be c² = a² + b².
    4. D)He incorrectly used sin(60°) instead of cos(60°) in the Cosine Rule formula.
    Show answer

    Answer: He incorrectly used sin(60°) instead of cos(60°) in the Cosine Rule formula.

    Hint: Carefully recall the exact form of the Cosine Rule. What trigonometric function is used?

  4. Q4.medium

    In triangle PQR, side p = 5 cm, side q = 8 cm, and the included angle R = 60°. Find the length of side r.
    1. A)√57 cm
    2. B)√64 cm
    3. C)7 cm
    4. D)√73 cm
    Show answer

    Answer: 7 cm

    Hint: Use the Cosine Rule to find a side when two sides and the included angle are known.

  5. Q5.medium

    A triangle has two sides of length 12 cm and 15 cm, and the included angle between them is 30°. What is the area of the triangle?
    1. A)90 cm²
    2. B)60 cm²
    3. C)45 cm²
    4. D)30 cm²
    Show answer

    Answer: 45 cm²

    Hint: The area of a triangle can be found using the formula 1/2 ab sin C, where a and b are sides and C is the included angle.

  6. Q6.medium

    From a point on the ground 50 meters away from the base of a building, the angle of elevation to the top of the building is 45°. What is the height of the building?
    1. A)50√3 m
    2. B)50 m
    3. C)25√3 m
    4. D)100 m
    Show answer

    Answer: 50 m

    Hint: Draw a right-angled triangle representing the situation. Consider which trigonometric ratio relates the opposite side (height) and adjacent side (distance).

  7. Q7.hard

    A triangle ABC has side a = 12 cm, side b = 15 cm, and angle A = 30°. How many distinct triangles can be formed with these measurements, and what are the possible values for angle B (to the nearest degree)?
    1. A)1 triangle; B ≈ 39°
    2. B)1 triangle; B ≈ 141°
    3. C)2 triangles; B ≈ 30°, 150°
    4. D)2 triangles; B ≈ 39°, 141°
    Show answer

    Answer: 2 triangles; B ≈ 39°, 141°

    Hint: Use the Sine Rule to find sinB. Remember that for any sine value, there are two possible angles in the range 0° to 180° that yield that value, and check if both form valid triangles.

  8. Q8.hard

    A plot of land is in the shape of a quadrilateral ABCD. The side lengths are AB = 80 m, BC = 100 m, CD = 90 m, and DA = 70 m. If angle ABC = 120°, find the total area of the quadrilateral ABCD to the nearest square metre.
    1. A)4500 m²
    2. B)4650 m²
    3. C)4812 m²
    4. D)5000 m²
    Show answer

    Answer: 4812 m²

    Hint: Divide the quadrilateral into two triangles. First, use the Cosine Rule to find the length of the diagonal connecting B and D (or A and C). Then, calculate the area of each triangle using the formula 1/2 ab sinC, and sum them up.

  9. Q9.hard

    A rectangular prism has dimensions length 12 cm, width 5 cm, and height 4 cm. Find the angle between the longest diagonal of the prism (from one corner to the opposite corner) and the base plane. Give your answer to one decimal place.
    1. A)17.1°
    2. B)18.4°
    3. C)20.3°
    4. D)22.6°
    Show answer

    Answer: 17.1°

    Hint: To find the angle between a line and a plane, consider the right-angled triangle formed by the diagonal, its projection onto the plane, and the height. You'll need Pythagoras' theorem to find the length of the projection.

These are 9 of the 60 questions available for Trigonometry Applications. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.