Basic Trigonometric Identities

The video begins by introducing reciprocal trigonometric identities, showing how sine, cosine, and tangent relate to their reciprocals (cosecant, secant, and cotangent, respectively). It uses a tangent-sine-cosine hexagon to demonstrate that the product of opposite functions equals one (e.g., sine * cosecant = 1), using various examples for clarity.

Next, quotient or ratio identities are explained. It highlights that the tangent of an angle is equal to the ratio of sine to cosine (tangent = sine/cosine) and provides proofs to validate these relationships by manipulating the reciprocal forms of the functions.

The video then explores product identities formed by adjacent functions on the hexagon. It proves that the product of two adjacent functions can result in the next function in the clockwise direction (e.g., sine * secant = tangent). Several examples are shown to confirm these relationships.

Pythagorean identities are introduced, derived from the hexagon and the unit circle equation (x^2 + y^2 = 1). It is demonstrated that sin^2(theta) + cos^2(theta) = 1, 1 + tan^2(theta) = sec^2(theta), and 1 + cot^2(theta) = csc^2(theta). Proofs are provided using a 30-degree angle example to validate these identities numerically.

The concept of even and odd trigonometric functions is discussed. Based on the symmetry of the unit circle, it is shown that cosine and secant are even functions (f(x) = f(-x)), while sine, tangent, cosecant, and cotangent are odd functions (f(-x) = -f(x)). This is illustrated with examples like cos(theta) = cos(-theta) and sin(theta) = -sin(-theta), using 60-degree and 150-degree angles.

Additional identities related to angles rotated by 90 degrees (theta + 90 degrees) are derived. These cofunction identities demonstrate relationships such as cos(theta + 90) = -sin(theta) and sin(theta + 90) = cos(theta). The derivations involve geometric interpretations on the unit circle and are verified using numerical examples like theta = 30 degrees.

Finally, identities for angles shifted by 180 degrees (theta +/- 180 degrees) are presented. It's shown that these angles are coterminal, leading to relationships like cos(theta +/- 180) = -cos(theta) and tan(theta +/- 180) = tan(theta). These identities are verified with a theta of 45 degrees, emphasizing their truthfulness for various angle measurements.