Sin(θ/2)

Definition

sin(θ/2) is a trigonometric function that represents the sine of half the angle θ. This term is particularly important in the context of double-angle, half-angle, and reduction formulas, as it allows for the simplification and manipulation of trigonometric expressions.

5 Must Know Facts For Your Next Test

1. The term $\sin(\theta/2)$ is commonly used in the derivation of half-angle formulas, which express the sine, cosine, or tangent of half an angle in terms of the original angle.
2. In the context of double-angle formulas, $\sin(\theta/2)$ is a key component in the expressions for $\sin(2\theta)$ and $\cos(2\theta)$.
3. Reduction formulas often involve the use of $\sin(\theta/2)$ to simplify trigonometric expressions, particularly when dealing with angles greater than 90 degrees.
4. The value of $\sin(\theta/2)$ is always positive, regardless of the quadrant in which the angle $\theta$ lies.
5. Understanding the properties and applications of $\sin(\theta/2)$ is crucial for effectively solving problems involving trigonometric identities and transformations.

Review Questions

• Explain how the term $\sin(\theta/2)$ is used in the derivation of half-angle formulas.
  • The term $\sin(\theta/2)$ is a key component in the derivation of half-angle formulas, which express the sine, cosine, or tangent of half an angle in terms of the original angle $\theta$. For example, the half-angle formula for the sine function is $\sin(\theta/2) = \pm\sqrt{(1 - \cos(\theta))/2}$. This formula allows us to simplify trigonometric expressions involving half-angles by using the known values of the original angle $\theta$.
• Describe the role of $\sin(\theta/2)$ in the double-angle formulas for $\sin(2\theta)$ and $\cos(2\theta)$.
  • The term $\sin(\theta/2)$ is a crucial component in the double-angle formulas for $\sin(2\theta)$ and $\cos(2\theta)$. The double-angle formula for the sine function is $\sin(2\theta) = 2\sin(\theta/2)\cos(\theta/2)$, and the double-angle formula for the cosine function is $\cos(2\theta) = \cos^2(\theta/2) - \sin^2(\theta/2)$. These formulas allow us to express the sine and cosine of twice an angle in terms of the sine and cosine of the original angle, which is particularly useful for simplifying and manipulating trigonometric expressions.
• Explain how the properties of $\sin(\theta/2)$ are leveraged in reduction formulas to simplify trigonometric expressions.
  • Reduction formulas often involve the use of $\sin(\theta/2)$ to simplify trigonometric expressions, particularly when dealing with angles greater than 90 degrees. This is because the value of $\sin(\theta/2)$ is always positive, regardless of the quadrant in which the angle $\theta$ lies. By expressing trigonometric functions in terms of $\sin(\theta/2)$, we can reduce the angle to a smaller, more manageable value and simplify the overall expression. This is especially useful when working with complex trigonometric identities or transformations that involve angles greater than 90 degrees.

Related terms

Double-Angle Formula: The double-angle formula is a trigonometric identity that expresses the sine, cosine, or tangent of twice an angle in terms of the sine, cosine, or tangent of the original angle.

Half-Angle Formula: The half-angle formula is a trigonometric identity that expresses the sine, cosine, or tangent of half an angle in terms of the sine, cosine, or tangent of the original angle.

Reduction Formula: Reduction formulas are trigonometric identities that allow for the simplification of trigonometric expressions by reducing the angle to a smaller, more manageable value.