GCSE Mathematics: Pythagoras and Trigonometry

GCSE Mathematics: Pythagoras and Trigonometry

In GCSE Mathematics, understanding Pythagoras' theorem and trigonometry is essential for solving problems involving right-angled triangles. This topic covers the application of Pythagoras' theorem in both 2D and 3D shapes, as well as the use of trigonometric ratios to find unknown angles and sides.

Pythagoras' Theorem

Pythagoras' theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):

c² = a² + b²

Worked Example

Problem: Find the length of the hypotenuse of a right-angled triangle where the other two sides are 3 cm and 4 cm.

Solution:

• Given: a = 3 cm, b = 4 cm
• Using Pythagoras' theorem: c² = 3² + 4²
• c² = 9 + 16 = 25
• c = √25 = 5 cm

Trigonometric Ratios

Trigonometry involves the relationships between the angles and sides of triangles. The primary ratios used in right-angled triangles are:

• Sine (sin): sin(θ) = opposite/hypotenuse
• Cosine (cos): cos(θ) = adjacent/hypotenuse
• Tangent (tan): tan(θ) = opposite/adjacent

These ratios can be remembered using the acronym SOHCAHTOA.

Worked Example

Problem: In a right-angled triangle, if the angle θ is 30° and the length of the hypotenuse is 10 cm, find the length of the opposite side.

Solution:

• Using the sine ratio: sin(30°) = opposite/10
• Since sin(30°) = 0.5, we have 0.5 = opposite/10
• Thus, opposite = 10 * 0.5 = 5 cm

Angles of Elevation and Depression

Angles of elevation and depression are also important concepts in trigonometry. The angle of elevation is the angle formed by the line of sight when looking up at an object, while the angle of depression is formed when looking down at an object. These angles can be solved using the same trigonometric ratios.

Higher Tier Content

For higher tier students, additional concepts include the Sine Rule, Cosine Rule, and the Area Rule for non-right-angled triangles:

• Sine Rule: a/sin(A) = b/sin(B) = c/sin(C)
• Cosine Rule: c² = a² + b² - 2ab cos(C)
• Area Rule: Area = 1/2ab sin(C)

Understanding these principles will enhance your ability to solve complex problems involving triangles in various contexts.