Trigonometric Functions of Any Angle - Unit Circle, Radians, Degrees, Coterminal & Reference Angles

The video begins by introducing the mnemonic SOH CAH TOA for sine, cosine, and tangent ratios in a right triangle. Sine (SOH) is opposite over hypotenuse, Cosine (CAH) is adjacent over hypotenuse, and Tangent (TOA) is opposite over adjacent. It then covers reciprocal identities: secant is 1/cosine, cosecant is 1/sine, and cotangent is 1/tangent. Examples are provided to calculate all six trigonometric functions for given right triangles.

The explanation continues with how to find a missing side of a right triangle using the Pythagorean theorem (a² + b² = c²). Special right triangles like 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41, and 11-60-61 are highlighted as useful shortcuts for quicker calculations. The concept of scaling these special triangles is also mentioned.

The video explains how to determine the quadrant an angle lies in based on the signs of sine, cosine, and tangent. Sine is positive in quadrants 1 and 2, cosine in 1 and 4, and tangent in 1 and 3. Examples demonstrate how to find the other five trigonometric functions when one function and the quadrant are given, using the Pythagorean theorem to find missing sides.

The process of finding all six trigonometric functions when given a terminal point (x, y) is detailed. This involves plotting the point, forming a right triangle, using the Pythagorean theorem to find the hypotenuse, and then calculating the trigonometric ratios. Rationalizing denominators is also covered in this section.

Practical applications of trigonometry are presented through word problems. These include finding missing sides or angles in real-world scenarios, such as calculating the height of a building using the angle of elevation or finding the length of a shadow using the angle of depression. The importance of choosing the correct trigonometric function (sine, cosine, or tangent) is emphasized.

The unit circle (a circle with radius 1) is introduced, explaining how sine corresponds to the y-value and cosine to the x-value in this context. The standard degree measures (0, 90, 180, 270, 360) and their radian equivalents (0, π/2, π, 3π/2, 2π) are outlined. The video provides instructions and examples for converting angles between degrees and radians.

The concept of coterminal angles is explained: angles that share the same terminal side. To find a positive coterminal angle, add 360° (or 2π radians), and to find a negative one, subtract 360° (or 2π radians). Coterminal angles have the same trigonometric values.

Reference angles, which are acute angles between the terminal side of an angle and the x-axis, are discussed. Formulas for calculating reference angles in each quadrant are provided. Examples include finding reference angles for both degree and radian measures, often by first converting radians to degrees for easier understanding.

This section focuses on evaluating trigonometric functions for common angles (30°, 45°, 60°) using the 30-60-90 and 45-45-90 special right triangles. The side ratios for these triangles are provided. The video then demonstrates how to use reference angles and quadrant rules to evaluate trigonometric functions for angles greater than 90° or for negative angles, both in degrees and radians.

The video concludes by applying the learned concepts to evaluate reciprocal trigonometric functions (secant, cosecant, cotangent) for various angles, including those greater than 360° and negative angles. The process involves finding the corresponding sine, cosine, or tangent value, determining its sign based on the quadrant, and then taking the reciprocal, often requiring rationalization of the denominator.