Sin(2θ)

Definition

The sine function of twice the angle θ, where θ is any real number. This trigonometric identity is important in the context of double-angle, half-angle, and reduction formulas, which allow for the simplification and manipulation of trigonometric expressions.

5 Must Know Facts For Your Next Test

1. The double-angle formula for sine states that $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.
2. The half-angle formula for sine states that $\sin(\theta/2) = \pm\sqrt{\frac{1-\cos(\theta)}{2}}$.
3. Reduction formulas allow trigonometric functions of large angles to be expressed in terms of functions of smaller angles, simplifying calculations.
4. The value of $\sin(2\theta)$ depends on the value of the original angle $\theta$, and can range from -1 to 1.
5. Knowing the properties of $\sin(2\theta)$ is crucial for solving problems involving double-angle, half-angle, and reduction formulas.

Review Questions

• Explain how the double-angle formula for sine, $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, is derived.
  • The double-angle formula for sine is derived by using the trigonometric identity $\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$. Substituting $A = \theta$ and $B = \theta$ into this identity, we get $\sin(2\theta) = \sin(\theta)\cos(\theta) + \cos(\theta)\sin(\theta)$. Factoring out $\sin(\theta)$ and $\cos(\theta)$ from the right-hand side, we arrive at the double-angle formula $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.
• Describe how the half-angle formula for sine, $\sin(\theta/2) = \pm\sqrt{\frac{1-\cos(\theta)}{2}}$, can be used to simplify trigonometric expressions.
  • The half-angle formula for sine allows us to express the sine of half an angle in terms of the cosine of the original angle. This is useful when we need to simplify trigonometric expressions involving half-angles. For example, if we have an expression containing $\sin(\theta/2)$, we can use the half-angle formula to rewrite it in terms of $\cos(\theta)$, which may be easier to evaluate or manipulate. This formula is particularly helpful in the context of double-angle and reduction formulas, where simplifying expressions involving half-angles is often required.
• Analyze the relationship between the values of $\sin(2\theta)$ and the original angle $\theta$, and explain how this understanding can be applied to solve problems.
  • The value of $\sin(2\theta)$ is directly related to the value of the original angle $\theta$. Since $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, the sign and magnitude of $\sin(2\theta)$ will depend on the values of $\sin(\theta)$ and $\cos(\theta)$. For example, when $\theta$ is in the first or second quadrant, $\sin(\theta)$ is positive, and the sign of $\sin(2\theta)$ will be determined by the sign of $\cos(\theta)$. Conversely, when $\theta$ is in the third or fourth quadrant, $\sin(\theta)$ is negative, and the sign of $\sin(2\theta)$ will be determined by the sign of $\cos(\theta)$. Understanding this relationship between $\sin(2\theta)$ and $\theta$ is crucial for solving problems involving double-angle, half-angle, and reduction formulas, as it allows you to predict the behavior of $\sin(2\theta)$ and use it to simplify or evaluate trigonometric expressions.

Related terms

Double-Angle Formula: A formula 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: A formula 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: A formula that expresses the sine, cosine, or tangent of an angle in terms of the sine, cosine, or tangent of a smaller angle.