EDEXCEL GCSE HIGHER MATHS - What are Trigonometry and Pythagoras Theorem

Edexcel GCSE Higher Maths: Trigonometry and Pythagoras Theorem

What is Trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used in many different fields, including engineering, architecture, and navigation.

Key Concepts:

• Right-angled triangles: A triangle with one angle measuring 90 degrees.
• Hypotenuse: The side opposite the right angle (always the longest side).
• Opposite: The side opposite the angle we are considering.
• Adjacent: The side next to the angle we are considering.
• Trigonometric Ratios:
  • Sine (sin): sin(angle) = Opposite / Hypotenuse
  • Cosine (cos): cos(angle) = Adjacent / Hypotenuse
  • Tangent (tan): tan(angle) = Opposite / Adjacent

SOH CAH TOA: This mnemonic is helpful for remembering the trigonometric ratios.

Example:

     /|
    / |
   /  |
  /   | Opposite
 /____|
  Adjacent
     Hypotenuse

Finding missing sides or angles:

• Finding a side: Use the appropriate trigonometric ratio and the given information.
• Finding an angle: Use the inverse trigonometric functions (sin-1, cos-1, tan-1).

Pythagoras Theorem

Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Formula:

• a2 + b2 = c2

where:

• a and b are the lengths of the two shorter sides
• c is the length of the hypotenuse

Example:

     /|
    / |
   /  |
  /   | 4 cm
 /____|
  3 cm
     5 cm

Finding a missing side:

1. Identify the hypotenuse (c).
2. Identify the two shorter sides (a and b).
3. Substitute the known values into the formula and solve for the unknown side.

Applications:

• Finding the length of a diagonal of a rectangle or square.
• Calculating the distance between two points.
• Solving problems in geometry and engineering.

Combining Trigonometry and Pythagoras Theorem

• Finding a side: You can use trigonometry to find one side of a right-angled triangle, then use Pythagoras Theorem to find another side.
• Finding an angle: You can use Pythagoras Theorem to find a missing side, then use trigonometry to find an angle.

Key Points to Remember

• Right-angled triangles are essential for both trigonometry and Pythagoras Theorem.
• Make sure you understand the definitions of opposite, adjacent, and hypotenuse.
• Always label the sides of the triangle correctly before applying the formulas.
• Practice using the formulas and apply them to real-life scenarios.

Practice Questions

1. In a right-angled triangle, the hypotenuse is 10 cm and one angle is 30 degrees. Find the length of the opposite side.
2. A ladder leans against a wall, making an angle of 70 degrees with the ground. The base of the ladder is 2 meters from the wall. How high up the wall does the ladder reach?
3. A rectangular garden is 12 meters long and 9 meters wide. What is the length of the diagonal path across the garden?

Remember to use your calculator to find the values of trigonometric functions and inverse functions.