Problem 24 - Algebra 2 and Trigonometry

Question

Show that \(\csc 2 \theta=\frac{1}{2} \sec \theta \csc \theta\)

Step-by-Step Solution

Answer

This identity is verified using sine and cosine definitions with double angle identities.

Step 1: Rewrite Secant and Cosecant in Terms of Sine and Cosine

Recall the reciprocal identities: \[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta}. \]Thus, we can express the given equation as:\[ \csc 2\theta = \frac{1}{2} \frac{1}{\cos \theta} \frac{1}{\sin \theta}. \]

Step 2: Use the Double Angle Identity for Sine

The sine double angle identity is:\[ \sin 2\theta = 2 \sin \theta \cos \theta. \]Therefore, the cosecant of double angle is:\[ \csc 2\theta = \frac{1}{\sin 2\theta} = \frac{1}{2 \sin \theta \cos \theta}. \]

Step 3: Compare Both Sides of the Equation

The right-hand side is:\[ \frac{1}{2} \sec \theta \csc \theta = \frac{1}{2} \cdot \frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} = \frac{1}{2 \sin \theta \cos \theta}. \]This matches the expression we found for the left-hand side in Step 2.

Step 4: Conclude the Verification

Both sides of the equation have been simplified to:\[ \csc 2\theta = \frac{1}{2 \sin \theta \cos \theta}. \]Therefore, the given identity is verified: \[ \csc 2\theta = \frac{1}{2} \sec \theta \csc \theta. \]

Key Concepts

Reciprocal Identities

In trigonometry, reciprocal identities are essential tools that help us understand the relationships between different trigonometric functions. Each of the primary trigonometric functions has a reciprocal counterpart:

• Secant (sec) is the reciprocal of cosine (cos): \[ \sec \theta = \frac{1}{\cos \theta} \]
• Cosecant (csc) is the reciprocal of sine (sin):\[ \csc \theta = \frac{1}{\sin \theta} \]
• Cotangent (cot) is the reciprocal of tangent (tan):\[ \cot \theta = \frac{1}{\tan \theta} \]

Understanding these reciprocal identities is crucial as they often simplify complex trigonometric expressions and equations. For example, when verifying the trigonometric identity \( \csc 2\theta=\frac{1}{2} \sec \theta \csc \theta \), we start by rewriting sec and csc using their reciprocal identities. This rewrite facilitates further simplifications and allows us to compare and verify the equation efficiently.

Double Angle Identity

The double angle identity is a special type of identity in trigonometry that helps us deal with trigonometric expressions involving angles that are doubled. One of the commonly used double angle identities is for the sine function:\[ \sin 2\theta = 2 \sin \theta \cos \theta \]This identity allows us to express the sine of double an angle, \( 2\theta \), in terms of the sine and cosine of the original angle, \( \theta \).
In the context of the problem, using the double angle identity for sine allows us to rewrite \( \csc 2\theta \) in a more manageable form:\[ \csc 2\theta = \frac{1}{\sin 2\theta} = \frac{1}{2 \sin \theta \cos \theta} \]This transformation simplifies the verification process by breaking the problem into smaller, more understandable components.

Trigonometric Functions

Understanding the basic trigonometric functions is fundamental in trigonometry. The six main trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. Each function relates to an angle in a right-angled triangle or the unit circle:

• Sine (sin): Ratio of the opposite side to the hypotenuse
• Cosine (cos): Ratio of the adjacent side to the hypotenuse
• Tangent (tan): Ratio of the opposite side to the adjacent side
• Cosecant (csc): Reciprocal of sine
• Secant (sec): Reciprocal of cosine
• Cotangent (cot): Reciprocal of tangent

These functions are not only vital for understanding geometric shapes and angles but also for solving trigonometric equations and identities. Being comfortable with these basic functions and their relationships via identities, such as reciprocal and double angle identities, enhances problem-solving efficiency and accuracy in trigonometry.