Tan(θ/2)

Definition

tan(θ/2) is a trigonometric function that represents the tangent 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 involving angles.

5 Must Know Facts For Your Next Test

1. The expression tan(θ/2) is often used in the derivation of double-angle and half-angle formulas for the trigonometric functions.
2. The half-angle formula for the tangent function is: tan(θ/2) = (1 - cos(θ)) / sin(θ).
3. tan(θ/2) can be used to simplify expressions involving trigonometric functions of angles that are multiples of the original angle.
4. The reduction formula for the tangent function is: tan(θ) = 2 tan(θ/2) / (1 - tan²(θ/2)).
5. Understanding the properties and applications of tan(θ/2) is crucial for solving a variety of trigonometric problems and manipulating trigonometric expressions.

Review Questions

• Explain how the expression tan(θ/2) is used in the derivation of double-angle and half-angle formulas.
  • The expression tan(θ/2) is central to the derivation of double-angle and half-angle formulas for the trigonometric functions. For example, the half-angle formula for the tangent function is derived using the identity tan(θ/2) = (1 - cos(θ)) / sin(θ). This formula allows for the simplification of expressions involving trigonometric functions of half an angle in terms of the original angle. Similarly, double-angle formulas can be derived using tan(θ/2) to express the trigonometric functions of a double angle in terms of the original angle.
• Describe how the reduction formula for the tangent function, tan(θ) = 2 tan(θ/2) / (1 - tan²(θ/2)), can be used to simplify trigonometric expressions.
  • The reduction formula for the tangent function, tan(θ) = 2 tan(θ/2) / (1 - tan²(θ/2)), can be used to simplify trigonometric expressions involving angles that are multiples of the original angle. By expressing the tangent of the larger angle in terms of tan(θ/2), the expression can be simplified and manipulated more easily. This formula is particularly useful when working with angles that are not easily expressed in terms of the standard reference angles (0°, 30°, 45°, 60°, 90°), as it allows for the reduction of the angle to a more manageable form.
• Analyze the relationship between tan(θ/2) and the other trigonometric functions, and explain how this relationship can be used to solve trigonometric equations and problems.
  • The expression tan(θ/2) is closely related to the other trigonometric functions, such as sine and cosine. The half-angle formula for the tangent function, tan(θ/2) = (1 - cos(θ)) / sin(θ), demonstrates this relationship. By understanding how tan(θ/2) is connected to sin(θ) and cos(θ), students can use this knowledge to solve a variety of trigonometric equations and problems. For example, if given an expression involving tan(θ/2), they can manipulate it using the half-angle formula to rewrite the expression in terms of sin(θ) and cos(θ), which may make the problem easier to solve. This relationship between tan(θ/2) and the other trigonometric functions is a powerful tool for simplifying and solving complex trigonometric expressions.

Related terms

Double-Angle Formulas: Formulas that express the trigonometric functions of a double angle in terms of the trigonometric functions of the original angle.

Half-Angle Formulas: Formulas that express the trigonometric functions of half an angle in terms of the trigonometric functions of the original angle.

Reduction Formulas: Formulas that express the trigonometric functions of an angle in terms of the trigonometric functions of a smaller angle.