Cos(θ/2)

Definition

The cosine function of half the angle θ, which is a trigonometric identity used in various formulas related to double-angle, half-angle, and reduction formulas in trigonometry.

5 Must Know Facts For Your Next Test

1. The term $\cos(\theta/2)$ is a key component of the half-angle formulas, which allow you to express the trigonometric functions of half an angle in terms of the original angle.
2. In the context of double-angle formulas, $\cos(\theta/2)$ is used to derive the expressions for $\cos(2\theta)$ and $\sin(2\theta)$.
3. Reduction formulas, such as those for the cosine and sine of an acute angle, also involve the term $\cos(\theta/2)$.
4. The value of $\cos(\theta/2)$ is always positive, as the cosine function is positive in the first and second quadrants of the unit circle.
5. Understanding the properties and applications of $\cos(\theta/2)$ is crucial for manipulating and simplifying trigonometric expressions involving double-angle, half-angle, and reduction formulas.

Review Questions

• Explain how the term $\cos(\theta/2)$ is used in the context of half-angle formulas.
  • The half-angle formulas express the sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the original angle. The term $\cos(\theta/2)$ is a key component of these formulas, as it allows you to calculate the cosine of half the angle $\theta$ based on the cosine of the full angle. For example, the half-angle formula for the cosine is $\cos(\theta/2) = \sqrt{(1 + \cos(\theta))/2}$, which demonstrates the central role of $\cos(\theta/2)$ in this identity.
• Describe how $\cos(\theta/2)$ is utilized in the double-angle formulas.
  • The double-angle formulas express the sine, cosine, and tangent of a double angle in terms of the trigonometric functions of the original angle. The term $\cos(\theta/2)$ appears in the derivation of these formulas. For instance, the double-angle formula for the cosine is $\cos(2\theta) = 2\cos^2(\theta/2) - 1$, which showcases how $\cos(\theta/2)$ is a crucial component in relating the cosine of the double angle to the cosine of the original angle.
• Analyze how the properties of $\cos(\theta/2)$ are leveraged in reduction formulas.
  • Reduction formulas allow you to express the trigonometric functions of an angle in terms of the functions of a related, smaller angle. The term $\cos(\theta/2)$ is employed in various reduction formulas, such as those for the cosine and sine of an acute angle. These formulas take advantage of the fact that $\cos(\theta/2)$ is always positive, as the cosine function is positive in the first and second quadrants of the unit circle. This property enables the simplification of trigonometric expressions by reducing the angle to a value between 0 and $\pi/2$.

Related terms

Double-Angle Formulas: Trigonometric identities that express the sine, cosine, and tangent of a double angle in terms of the sine, cosine, and tangent of the original angle.

Half-Angle Formulas: Trigonometric identities that express the sine, cosine, and tangent of half an angle in terms of the sine, cosine, and tangent of the original angle.

Reduction Formulas: Trigonometric identities that express the sine, cosine, and tangent of an angle in terms of the sine, cosine, and tangent of a related, smaller angle.