Are You a Trigonometry Pro?

Trigonometry: Relationships between angles and ratios of lengths

• Trigonometry is a branch of mathematics concerned with relationships between angles and ratios of lengths.

• The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

• Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.

• Trigonometry is known for its many identities, which are commonly used for rewriting trigonometrical expressions to simplify an expression, find a more useful form of an expression, or solve an equation.

• Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.

• In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems equivalent to modern trigonometric formulae.

• The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios such as sine.

• The modern sine convention is first attested in the Surya Siddhanta, and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer Aryabhata.

• Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.

• Trigonometric ratios are the ratios between edges of a right triangle, which define functions of this angle that are called trigonometric functions.

• Trigonometric functions were among the earliest uses for mathematical tables.

• Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.Trigonometry: Laws, Formulas, and Identities

• The Law of Cosines extends the Pythagorean theorem to arbitrary triangles.

• The Law of Tangents is an alternative to the Law of Cosines for solving unknown edges of a triangle.

• The area of a triangle can be calculated using the product of two sides and the sine of the angle between them.

• Heron's formula can also be used to calculate the area of a triangle.

• Pythagorean identities are related to the Pythagorean theorem and hold for any value.

• Euler's formula produces analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i.

• Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.