UK K12 GCSE/A-Level: Year 10 GCSE Mathematics - Trigonometry, Non-Right Triangles, Sine/Cosine Rule

Learning Objectives

By the end of this topic, students will be able to:

• Apply the Sine Rule to find unknown side lengths in non-right-angled triangles.
• Apply the Cosine Rule to find unknown side lengths in non-right-angled triangles.
• Use the Sine Rule and Cosine Rule to solve problems involving the height of an object or the distance between two points.
• Understand the conditions under which the Sine Rule and Cosine Rule can be applied.
• Use trigonometric identities to simplify expressions and solve equations.

Core Concepts

The Sine Rule

The Sine Rule is a formula used to find the length of a side in a non-right-angled triangle when the lengths of the other two sides and the sine of one of the angles are known. The formula is:

a / sin(A) = b / sin(B) = c / sin(C)

where a, b, and c are the lengths of the sides opposite angles A, B, and C respectively.

The Cosine Rule

The Cosine Rule is a formula used to find the length of a side in a non-right-angled triangle when the lengths of the other two sides and the cosine of one of the angles are known. The formula is:

c² = a² + b² - 2ab * cos(C)

where a, b, and c are the lengths of the sides opposite angles A, B, and C respectively.

Conditions for the Sine Rule and Cosine Rule

The Sine Rule can be applied to any triangle, but the Cosine Rule can only be applied to triangles where all the sides are known.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables. They can be used to simplify expressions and solve equations.

Real-World Applications

The Sine Rule and Cosine Rule have many real-world applications, such as:

• Calculating the height of a building or a tree
• Finding the distance between two points on the Earth's surface
• Determining the length of a shadow

Worked Examples

Example 1: Using the Sine Rule

In triangle ABC, angle A = 60°, angle B = 30°, and side a = 10cm. Find the length of side b.

Using the Sine Rule, we have:

b / sin(B) = a / sin(A) b / sin(30°) = 10 / sin(60°) b = 10 * sin(30°) / sin(60°) b = 10 * 0.5 / 0.866 b = 5.77cm

Example 2: Using the Cosine Rule

In triangle ABC, side a = 10cm, side b = 12cm, and angle C = 60°. Find the length of side c.

Using the Cosine Rule, we have:

c² = a² + b² - 2ab * cos(C) c² = 10² + 12² - 2 * 10 * 12 * cos(60°) c² = 100 + 144 - 240 * 0.5 c² = 244 - 120 c² = 124 c = ?124 c = 11.13cm

Common Misconceptions

• Many students believe that the Sine Rule and Cosine Rule can only be applied to right-angled triangles. This is not true - they can be applied to any triangle.
• Some students think that the Sine Rule and Cosine Rule are only used to find the length of a side. This is not true - they can also be used to find the height of an object or the distance between two points.
• A common mistake is to use the Sine Rule and Cosine Rule in the wrong order. Make sure to use the correct formula for the given information.

Exam Tips

• Make sure to read the question carefully and identify the type of triangle (right-angled or non-right-angled).
• Use the correct formula for the given information (Sine Rule or Cosine Rule).
• Check your units and make sure they are consistent.
• Use trigonometric identities to simplify expressions and solve equations.
• Practice, practice, practice - the more you practice, the more confident you will become.

MCQs

Question 1: [F] Using the Sine Rule

In triangle ABC, angle A = 30°, angle B = 60°, and side a = 8cm. Find the length of side b.

A) 6cm B) 8cm C) 10cm D) 12cm

Correct answer: B) 8cm Why the distractors fail: A) This is too small - the Sine Rule would give a larger value. C) This is too large - the Sine Rule would give a smaller value. D) This is not possible - the Sine Rule would give a different value.

Question 2: [H] Using the Cosine Rule

In triangle ABC, side a = 12cm, side b = 15cm, and angle C = 60°. Find the length of side c.

A) 10cm B) 12cm C) 15cm D) 18cm

Correct answer: D) 18cm Why the distractors fail: A) This is too small - the Cosine Rule would give a larger value. B) This is too small - the Cosine Rule would give a larger value. C) This is too small - the Cosine Rule would give a larger value.

Question 3: [F] Conditions for the Sine Rule and Cosine Rule

Which of the following is a condition for using the Sine Rule?

A) All sides must be known B) All angles must be known C) Any two sides and the sine of the included angle must be known D) Any two sides and the cosine of the included angle must be known

Correct answer: C) Any two sides and the sine of the included angle must be known Why the distractors fail: A) This is a condition for using the Cosine Rule, not the Sine Rule. B) This is not a condition for using the Sine Rule. D) This is not a condition for using the Sine Rule.

Question 4: [H] Trigonometric Identities

Simplify the expression: sin(2?) = 2sin(?)cos(?)

A) sin(?) + cos(?) B) 2sin(?)cos(?) C) sin(?) - cos(?) D) 2cos(?) - sin(?)

Correct answer: B) 2sin(?)cos(?) Why the distractors fail: A) This is not the correct simplification. C) This is not the correct simplification. D) This is not the correct simplification.

Question 5: [F] Real-World Applications

What is a real-world application of the Sine Rule?

A) Calculating the area of a triangle B) Finding the height of a building C) Determining the length of a shadow D) Calculating the volume of a pyramid

Correct answer: B) Finding the height of a building Why the distractors fail: A) This is not a real-world application of the Sine Rule. C) This is not a real-world application of the Sine Rule. D) This is not a real-world application of the Sine Rule.

Short-Answer Questions

Question 1

Explain the difference between the Sine Rule and the Cosine Rule.

Question 2

Use the Sine Rule to find the length of side b in triangle ABC, where angle A = 60°, angle B = 30°, and side a = 10cm.

Question 3

Use the Cosine Rule to find the length of side c in triangle ABC, where side a = 12cm, side b = 15cm, and angle C = 60°.

Question 4

Simplify the expression: sin(2?) = 2sin(?)cos(?)

Question 5

Explain a real-world application of the Sine Rule.