Half-Angle Identities in Trigonometry

In trigonometry, half-angle identities are a set of formulas that express trigonometric functions of half an angle in terms of trigonometric functions of the full angle. These identities are incredibly useful for simplifying trigonometric expressions, solving equations, and finding exact values for trigonometric functions.

Derivation of the Half-Angle Identities

The half-angle identities can be derived from the double-angle formulas. Let's consider the double-angle formula for cosine:

cos 2θ = 1 - 2sin²θ

Solving for sin²θ, we get:

sin²θ = (1 - cos 2θ) / 2

Now, let θ = x/2. Substituting this into the equation above, we get the half-angle identity for sine:

sin²(x/2) = (1 - cos x) / 2

Taking the square root of both sides, we get:

sin(x/2) = ±√((1 - cos x) / 2)

The sign of the square root depends on the quadrant of x/2.

Similarly, we can derive the half-angle identities for cosine and tangent using the double-angle formulas:

Half-Angle Identity for Cosine

cos²(x/2) = (1 + cos x) / 2

cos(x/2) = ±√((1 + cos x) / 2)

Half-Angle Identity for Tangent

tan(x/2) = sin x / (1 + cos x)

Applications of Half-Angle Identities

Here are some key applications of half-angle identities:

• Simplifying Trigonometric Expressions: Half-angle identities can be used to simplify complex trigonometric expressions by replacing functions of half angles with functions of whole angles.
• Solving Trigonometric Equations: These identities can help solve trigonometric equations by reducing the angle in the equation and making it easier to find solutions.
• Finding Exact Values: Half-angle identities are crucial for finding exact values of trigonometric functions for angles that are not standard angles (like 30°, 45°, 60°).

Example: Finding the Exact Value of sin(15°):

Using the half-angle identity for sine, we can find the exact value of sin(15°):

sin(15°) = sin(30°/2) = ±√((1 - cos 30°) / 2)

Since 15° is in the first quadrant, sin(15°) is positive. We know cos 30° = √3/2. Substituting, we get:

sin(15°) = √((1 - √3/2) / 2) = √(2 - √3) / 2

Conclusion

Half-angle identities are powerful tools in trigonometry. They provide a way to relate trigonometric functions of half angles to functions of whole angles, simplifying expressions, solving equations, and finding exact values. Mastering these identities is essential for a deeper understanding of trigonometry and its applications.