The half-angle formula is a mathematical formula used to find the trigonometric values of half angles. It is particularly useful when dealing with trigonometric functions of angles that are difficult to evaluate directly.
The half-angle formula contains the following knowledge points:
The half-angle formula can be expressed as follows:
For sine: sin(x/2) = ±√[(1 - cos(x))/2]
For cosine: cos(x/2) = ±√[(1 + cos(x))/2]
For tangent: tan(x/2) = ±√[(1 - cos(x))/(1 + cos(x))]
To apply the half-angle formula, follow these steps:
The symbol for the half-angle formula is "x/2", where x represents the original angle.
There are several methods to derive the half-angle formula, including:
Find the value of sin(45°/2). Solution: Using the half-angle formula for sine: sin(45°/2) = ±√[(1 - cos(45°))/2] Since cos(45°) = √2/2, we have: sin(45°/2) = ±√[(1 - √2/2)/2] Simplifying further, we get: sin(45°/2) = ±√[(2 - √2)/4]
Find the value of cos(60°/2). Solution: Using the half-angle formula for cosine: cos(60°/2) = ±√[(1 + cos(60°))/2] Since cos(60°) = 1/2, we have: cos(60°/2) = ±√[(1 + 1/2)/2] Simplifying further, we get: cos(60°/2) = ±√[(3/2)/2]
Q: What is the half-angle formula? The half-angle formula is a mathematical formula used to find the trigonometric values of half angles.
Q: How do I apply the half-angle formula? To apply the half-angle formula, you need to determine the value of the angle for which you want to find the half-angle, and then substitute it into the respective formula.
Q: Are there different formulas for different trigonometric functions? Yes, there are separate formulas for sine, cosine, and tangent in the half-angle formula.
Q: Can the half-angle formula be derived from other trigonometric identities? Yes, the half-angle formula can be derived using the double-angle formula and Pythagorean identities, among other methods.