UK K12 GCSE/A-Level: Year 9 KS3/Pre-GCSE Mathematics - Trigonometry, Sin, Cos, Tan in Right-Angled Triangles

Learning Objectives

By the end of this topic, students will be able to: - Define sin, cos, and tan in the context of right-angled triangles. - Identify the relationships between the angles and side lengths of a right-angled triangle. - Apply the sin, cos, and tan ratios to solve problems involving right-angled triangles. - Recognize and explain the limitations of trigonometric ratios in solving problems.

Core Concepts

In a right-angled triangle, sin, cos, and tan are defined as ratios of the side lengths. The sin of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cos of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The tan of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

These ratios can be expressed mathematically as:

• sin(?) = opposite side / hypotenuse
• cos(?) = adjacent side / hypotenuse
• tan(?) = opposite side / adjacent side

Understanding the relationships between the angles and side lengths of a right-angled triangle is crucial in applying trigonometric ratios. The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Key Concepts: Angles and Side Lengths

• The hypotenuse is the side opposite the right angle.
• The opposite side is the side opposite the angle being considered.
• The adjacent side is the side adjacent to the angle being considered.

Key Concepts: Trigonometric Ratios

• sin(?) is the ratio of the opposite side to the hypotenuse.
• cos(?) is the ratio of the adjacent side to the hypotenuse.
• tan(?) is the ratio of the opposite side to the adjacent side.

Worked Examples

Example 1: Finding sin, cos, and tan

In a right-angled triangle, the length of the hypotenuse is 10 cm and the length of the side opposite the angle is 6 cm. Find the sin, cos, and tan of the angle.

• sin(?) = opposite side / hypotenuse = 6 / 10 = 0.6
• cos(?) = adjacent side / hypotenuse = ?(hypotenuse^2 - opposite side^2) / hypotenuse = ?(100 - 36) / 10 = ?64 / 10 = 0.8
• tan(?) = opposite side / adjacent side = 6 / (?(100 - 36)) = 6 / (?64) = 6 / 8 = 0.75

Example 2: Solving a Problem

In a right-angled triangle, the length of the side opposite the angle is 8 cm and the length of the adjacent side is 6 cm. Find the length of the hypotenuse.

• tan(?) = opposite side / adjacent side = 8 / 6 = 1.33
• tan(?) = opposite side / adjacent side = 8 / 6 = 1.33
• sin(?) = opposite side / hypotenuse = 8 / hypotenuse
• cos(?) = adjacent side / hypotenuse = 6 / hypotenuse
• Using the Pythagorean theorem, we can find the length of the hypotenuse: hypotenuse^2 = opposite side^2 + adjacent side^2 = 8^2 + 6^2 = 64 + 36 = 100
• hypotenuse = ?100 = 10 cm

Common Misconceptions

• Misconception 1: Students may think that sin, cos, and tan are only used to find the length of the hypotenuse.
• Misconception 2: Students may think that sin, cos, and tan are only used to find the length of the opposite side.
• Misconception 3: Students may think that sin, cos, and tan are only used to find the length of the adjacent side.

Exam Tips

• Make sure to read the question carefully and identify what is being asked.
• Use the Pythagorean theorem to find the length of the hypotenuse.
• Use the trigonometric ratios to find the length of the opposite side or the adjacent side.
• Check your answers to make sure they are reasonable.

MCQs with Explanations

Question 1: [F]

What is the sin of an angle in a right-angled triangle?

A) The ratio of the adjacent side to the hypotenuse B) The ratio of the opposite side to the hypotenuse C) The ratio of the hypotenuse to the adjacent side D) The ratio of the hypotenuse to the opposite side

Correct answer: B) The ratio of the opposite side to the hypotenuse Why the distractors fail: A) This is the definition of cos. C) This is not a valid definition of sin. D) This is not a valid definition of sin.

Question 2: [H]

In a right-angled triangle, the length of the hypotenuse is 10 cm and the length of the side opposite the angle is 6 cm. What is the cos of the angle?

A) 0.6 B) 0.8 C) 1.33 D) 1.67

Correct answer: B) 0.8 Why the distractors fail: A) This is the value of sin. C) This is the value of tan. D) This is not a valid value for cos.

Question 3: [F]

What is the tan of an angle in a right-angled triangle?

A) The ratio of the opposite side to the adjacent side B) The ratio of the adjacent side to the opposite side C) The ratio of the hypotenuse to the adjacent side D) The ratio of the hypotenuse to the opposite side

Correct answer: A) The ratio of the opposite side to the adjacent side Why the distractors fail: B) This is not a valid definition of tan. C) This is not a valid definition of tan. D) This is not a valid definition of tan.

Question 4: [H]

In a right-angled triangle, the length of the side opposite the angle is 8 cm and the length of the adjacent side is 6 cm. What is the length of the hypotenuse?

A) 5 cm B) 6 cm C) 8 cm D) 10 cm

Correct answer: D) 10 cm Why the distractors fail: A) This is not a valid value for the length of the hypotenuse. B) This is the value of the adjacent side. C) This is the value of the opposite side.

Question 5: [F]

What is the relationship between the sin, cos, and tan ratios in a right-angled triangle?

A) sin = cos = tan B) sin = cos / tan C) sin / cos = tan D) sin / cos = 1 / tan

Correct answer: C) sin / cos = tan Why the distractors fail: A) This is not a valid relationship between the ratios. B) This is not a valid relationship between the ratios. D) This is not a valid relationship between the ratios.

Short-answer Questions

Question 1

In a right-angled triangle, the length of the hypotenuse is 10 cm and the length of the side opposite the angle is 6 cm. Find the sin, cos, and tan of the angle.

Question 2

In a right-angled triangle, the length of the side opposite the angle is 8 cm and the length of the adjacent side is 6 cm. Find the length of the hypotenuse.

Question 3

What is the relationship between the sin, cos, and tan ratios in a right-angled triangle?