Fill in the blank to complete the trigonometric formula. (Select all that apply.) sin u 2 = __
The Correct Answer and Explanation is:
To complete the trigonometric identity, we analyze the expression:
sin(u/2) = ______
This refers to the half-angle identity in trigonometry. The correct formula is:sin(u2)=±1−cosu2\sin\left(\frac{u}{2}\right) = \pm \sqrt{\frac{1 – \cos u}{2}}sin(2u)=±21−cosu
✅ Correct answers:
- sin(u2)=±1−cosu2\sin\left(\frac{u}{2}\right) = \pm \sqrt{\frac{1 – \cos u}{2}}sin(2u)=±21−cosu
📘 Explanation (300 words):
The half-angle identities are derived from the cosine and sine double-angle identities and are useful for expressing trigonometric functions of half angles (e.g., u2\frac{u}{2}2u) in terms of the full angle uuu. In particular, the half-angle identity for sine is:sin(u2)=±1−cosu2\sin\left(\frac{u}{2}\right) = \pm \sqrt{\frac{1 – \cos u}{2}}sin(2u)=±21−cosu
This identity is obtained by rearranging the cosine double-angle identity:cos(2θ)=1−2sin2(θ)\cos(2\theta) = 1 – 2\sin^2(\theta)cos(2θ)=1−2sin2(θ)
Solving for sin(θ)\sin(\theta)sin(θ), we get:sin2(θ)=1−cos(2θ)2\sin^2(\theta) = \frac{1 – \cos(2\theta)}{2}sin2(θ)=21−cos(2θ)
Then, replacing θ\thetaθ with u2\frac{u}{2}2u, we find:sin2(u2)=1−cosu2\sin^2\left(\frac{u}{2}\right) = \frac{1 – \cos u}{2}sin2(2u)=21−cosu
Taking the square root of both sides gives:sin(u2)=±1−cosu2\sin\left(\frac{u}{2}\right) = \pm \sqrt{\frac{1 – \cos u}{2}}sin(2u)=±21−cosu
The ± (plus-minus) sign depends on the quadrant in which u2\frac{u}{2}2u lies. For example:
- If u2\frac{u}{2}2u is in Quadrant I or II, then sine is positive.
- If u2\frac{u}{2}2u is in Quadrant III or IV, then sine is negative.
This identity is especially useful in solving trigonometric equations, evaluating angles not commonly found on the unit circle, or integrating trigonometric expressions. Understanding this identity provides a powerful tool in both pure and applied mathematics, especially in calculus and physics where angular functions often need to be rewritten in more convenient forms.
