Problem 59 - Precalculus Essentials

Question

In Exercises \(59-68\), verify each identity. $$\sin ^{2} \frac{\theta}{2}=\frac{\sec \theta-1}{2 \sec \theta}$$

Step-by-Step Solution

Answer

Hence, \( \sin ^{2} \frac{\theta}{2}=\frac{\sec \theta-1}{2 \sec \theta} \) is proven true.

Step 1: Express secant in terms of cosine

Replace \( sec \theta \) with \( \frac{1}{cos \theta} \). This gives us: \( \sin ^{2} \frac{\theta}{2}=\frac{\frac{1}{\cos \theta}-1}{2 * \frac{1}{\cos \theta}} \)

Step 2: Simplify the equation

By simplifying, we get \( \sin ^{2} \frac{\theta}{2}=\frac{1-\cos \theta}{2} \).

Step 3: Apply the half-angle identity

We know from the half-angle identity that \( \sin ^{2} \frac{\theta}{2}=\frac{1-\cos \theta}{2} \). The identity holds true, confirming that it's correct.

Key Concepts

Half-Angle Identities

In trigonometry, half-angle identities are useful for expressing trigonometric functions of angles that are half the size of a given angle. These identities are particularly valuable because they help derive values for angles that are often not directly available through the basic trigonometric table. The key is that they relate functions of angles to functions of half-angles.
For example, the half-angle identity for sine states that \[\sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2}\]This identity simplifies the process of working with expressions or equations that involve \(\theta\) and allows us to break down complex expressions into manageable components. By leveraging half-angle identities, you can tackle trigonometric equations that might initially seem daunting.
The use of half-angle identities is sometimes required in precalculus courses, where students are working to understand and verify trigonometric equations, much like the example exercise analyzed here.

Trigonometric Functions

Trigonometric functions are the fundamental building blocks in trigonometry. The primary trigonometric functions—sine, cosine, and tangent—are defined using the ratios of the sides of a right triangle or the points on the unit circle.

• **Sine** \(\sin\theta\) measures the ratio of the length of the opposite side to the hypotenuse.
• **Cosine** \(\cos\theta\) measures the ratio of the adjacent side to the hypotenuse.
• **Tangent** \(\tan\theta\) represents the ratio of the opposite side to the adjacent side.

Secondary functions include secant, cosecant, and cotangent, which are reciprocals of the primary functions.
In this exercise, secant, denoted as \(\sec\theta = \frac{1}{\cos\theta}\), is the reciprocal of cosine and often appears in trigonometric identities. Understanding how to manipulate these functions and their relationships is critical in verifying trigonometric identities and solving equations, as illustrated in the original exercise. Secant, in particular, helps to connect the dots between angle measures and function values, aiding in the verification of half-angle identities.

Precalculus

Precalculus serves as the bridge between algebra and calculus, exploring mathematical concepts more advanced than basic algebra but necessary for calculus. Topics in precalculus often include a study of functions and their properties, transformations, and manipulations.
A significant portion of precalculus involves understanding trigonometric functions and identities deeply. Students in precalculus learn to manipulate and verify trigonometric identities—like the half-angle identities—as crucial tools for simplifying and solving equations.

• Understanding how to verify identities: Involves substituting equivalent expressions and using known identities to prove that two sides of an equation are equal.
• Simplifying expressions: Through practice, students learn to express trigonometric functions in different, often simpler forms.
• Building a foundation for calculus: Adequately grasping these concepts prepares students for limits, derivatives, and integrals in calculus by familiarizing them with core functions and their properties.

In the example exercise, effectively applying these precalculus skills is essential for successfully verifying the given trigonometric identity.