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Trigonometry and Pythagoras' Theorem

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Trigonometry is a branch of mathematics that deals with the relationships between triangle angles and sides, especially in right-angled triangles. It builds on Pythagoras' theorem, enabling the calculation of unknown angles and sides using trigonometric ratios: sine, cosine, and tangent. These ratios, encapsulated by the mnemonic SOHCAHTOA, are fundamental for solving mathematical problems involving triangles. The text also covers the use of inverse trigonometric functions to find missing angles and the importance of memorizing key values for manual problem-solving.

Fundamentals of Trigonometry

Trigonometry is a fundamental branch of mathematics that explores the relationships between the angles and sides of triangles, with a focus on right-angled triangles. It extends beyond the scope of Pythagoras' theorem, which is limited to calculating the lengths of sides in right-angled triangles, by providing methods to determine unknown angles and sides. The core of trigonometry lies in the trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). These ratios are crucial for relating the angles of a triangle to the lengths of its sides and form the backbone of solving a multitude of mathematical problems that involve triangular shapes.
Right-angled triangle formed by three wooden rulers on a desk, with a clear protractor and a compass set for drawing, casting soft shadows.

Pythagoras' Theorem as a Foundation for Trigonometry

Pythagoras' theorem is a critical stepping stone in the journey towards understanding trigonometry. It 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 other two sides (a and b), expressed as a² + b² = c². This theorem is instrumental in finding the length of a missing side in a right-angled triangle when the lengths of the other two sides are known. While Pythagoras' theorem does not account for angles, it sets the stage for trigonometry, which expands the mathematical toolkit to include angle-related calculations in any type of triangle.

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00

Trigonometric Ratios Definition

Sine, cosine, and tangent are functions relating angles to side ratios in right-angled triangles.

01

Application of Trigonometry Beyond Right-Angled Triangles

Trigonometry extends to non-right-angled triangles using laws of sines and cosines to solve problems.

02

Pythagoras' Theorem Limitation

Pythagoras' theorem only calculates lengths of sides in right-angled triangles, not angles or sides in other triangles.

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