The term θ/2 refers to the angle that is half the value of the original angle θ. It is a fundamental concept in the context of double-angle, half-angle, and reduction formulas, which are used to simplify and manipulate trigonometric expressions.
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The angle θ/2 is used extensively in the derivation and application of double-angle, half-angle, and reduction formulas for trigonometric functions.
The half-angle formulas, such as $\sin(\theta/2) = \pm\sqrt{(1-\cos(\theta))/2}$ and $\cos(\theta/2) = \pm\sqrt{(1+\cos(\theta))/2}$, rely on the angle θ/2.
Double-angle formulas, such as $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ and $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$, can be derived using the half-angle formulas and the angle θ/2.
Reduction formulas, which express trigonometric functions of large angles in terms of those of smaller angles, often involve the angle θ/2 as an intermediate step.
The angle θ/2 is a key concept in understanding the relationships between trigonometric functions and their transformations, which is essential for solving a variety of trigonometric problems.
Review Questions
Explain how the angle θ/2 is used in the derivation of half-angle formulas for trigonometric functions.
The angle θ/2 is central to the derivation of half-angle formulas, such as $\sin(\theta/2) = \pm\sqrt{(1-\cos(\theta))/2}$ and $\cos(\theta/2) = \pm\sqrt{(1+\cos(\theta))/2}$. These formulas express the trigonometric functions of an angle with half the value of the original angle θ in terms of the original angle. By using the angle θ/2 as an intermediate step, these half-angle formulas can be derived from the fundamental trigonometric identities and the properties of the unit circle.
Describe how the angle θ/2 is used in the derivation of double-angle formulas for trigonometric functions.
The angle θ/2 plays a crucial role in the derivation of double-angle formulas, such as $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ and $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$. These formulas express the trigonometric functions of an angle with twice the value of the original angle θ in terms of the original angle. By first expressing the trigonometric functions of the angle 2θ in terms of the angle θ/2 using the half-angle formulas, and then simplifying the resulting expressions, the double-angle formulas can be obtained.
Analyze how the angle θ/2 is used in the context of reduction formulas for trigonometric functions.
Reduction formulas are used to express the trigonometric functions of large angles in terms of those of smaller angles. The angle θ/2 is often an intermediate step in the derivation of these formulas. By first expressing the trigonometric functions of the large angle in terms of the angle θ/2 using the half-angle formulas, and then further simplifying the expressions using the fundamental trigonometric identities, the reduction formulas can be obtained. This process allows for the simplification of complex trigonometric expressions and the efficient calculation of trigonometric values for large angles.