GCSE Mathematics: Pythagoras and Trigonometry

GCSE Mathematics: Pythagoras and Trigonometry In GCSE Mathematics, understanding Pythagoras' theorem and trigonometry is essential for solving problems involvin...

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 applying Pythagoras' theorem in both 2D and 3D shapes to find unknown sides, as well as using the sine, cosine, and tangent ratios (collectively known as SOHCAHTOA) to determine angles and sides in right-angled triangles.

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
c² = 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 used to find unknown angles or sides when certain values are known.

Worked Example

Problem: In a right-angled triangle, if one angle is 30° and the adjacent side is 5 cm, find the length of the opposite side.

Solution:

Using the tangent ratio: tan(30°) = opposite/5
tan(30°) = 1/√3 (approximately 0.577)
0.577 = opposite/5
opposite = 5 * 0.577 ≈ 2.89 cm

Angles of Elevation and Depression

In real-world applications, angles of elevation and depression are crucial. 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.

Higher Tier Concepts

For higher-tier students, additional concepts include:

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)

These rules allow for solving problems involving non-right-angled triangles and calculating areas.

Understanding these concepts is vital for success in GCSE Mathematics and provides a foundation for further studies in mathematics and related fields.