Use the unit-circle definitions of sine and cosine to prove...

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another and hold true for all valid angles. The identity sin² ? + cos² ? = 1, derived using the unit circle and the Pythagorean theorem, is one of the most fundamental identities in trigonometry. It forms the basis for many other results and applications within the subject.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. When applied to the unit circle, where the hypotenuse is the radius (equal to one), this theorem directly leads to the identity cos² ? + sin² ? = 1.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of the coordinate plane. It is fundamental in trigonometry because it provides a geometric framework for defining the sine and cosine functions, where any point on the circle is represented as (cos ?, sin ?). This allows for a clear, visual interpretation of these functions based on the coordinates of points along the circle.

Sine and Cosine

Sine and cosine are core trigonometric functions that can be defined using the unit circle. For any angle ? measured from the positive x-axis, the coordinates (cos ?, sin ?) of the point where the terminal side of the angle intersects the unit circle give the values of cosine and sine, respectively. This definition links algebraic and geometric interpretations of these functions.

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