91Ó°ÊÓ

Understanding half-angle formulas is crucial for establishing various trigonometric identities. These formulas express trigonometric functions of half-angles in terms of the trigonometric functions of the original angle. This becomes especially useful when solving complex trigonometric expressions and proofs.
For sine and cosine, the half-angle identities are:

• \(\text{sin} \frac{v}{2} = \text{sqrt} \frac{1 - \text{cos}(v)}{2}\)
• \(\text{cos} \frac{v}{2} = \text{sqrt} \frac{1 + \text{cos}(v)}{2}\)

To derive these formulas, one typically starts with the double-angle identities and manipulates them algebraically.
For example, \(\text{cos}(2u) = 2 \text{cos}^2(u) - 1\) can be rewritten to \(\text{cos}(u) = \text{sqrt} \frac{1 + \text{cos}(2u)}{2}\). Similarly, half-angle formulas help simplify many trigonometric problems efficiently.

The tangent of half an angle can also be expressed using half-angle formulas. Specifically, it can be rephrased as: \(\tan \frac{v}{2} = \frac{\text{sin}(\frac{v}{2})}{\text{cos}(\frac{v}{2})}\).
Using the half-angle identities for sine and cosine mentioned above, we substitute to get: \(\tan \frac{v}{2} = \frac{\text{sqrt}\frac{1 - \text{cos}(v)}{2}}{\text{sqrt}\frac{1 + \text{cos}(v)}{2}}\).
By simplifying the expression, we arrive at: \(\tan \frac{v}{2} = \text{sqrt}\frac{1 - \text{cos}(v)}{1 + \text{cos}(v)}\).
This formula is beneficial for problems that require manipulating angles within the range of 0 to \(2\pi\), enabling easier simplifications of trigonometric identities and equations.

The definitions of sine and cosine are fundamental to understanding and using trigonometric identities. In a right-angled triangle:

• \(\text{sin}(v) = \frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\text{cos}(v) = \frac{\text{adjacent}}{\text{hypotenuse}}\)

These basic definitions extend to the unit circle, where the angle is measured from the positive x-axis:

• \(\text{sin}(v)\) corresponds to the y-coordinate of the point on the unit circle.
• \(\text{cos}(v)\) corresponds to the x-coordinate.

Additionally, the co-function identities relate sine and cosine: \(\text{sin}(90^\text{°} - v) = \text{cos}(v)\) and \(\text{cos}(90^\text{°} - v) = \text{sin}(v)\). Understanding these definitions and their relationships is essential for smooth navigation through more complex trigonometric problems and proving identities.