Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use trigonometric identities to transform one side of the equation into the other \((0<{\theta}<\pi /2)\). $$\csc \theta \tan \theta=\sec \theta$$

Step-by-Step Solution

Answer

After using reciprocal and quotient identities, we can transform \( \csc \theta \tan \theta \) into \( \sec \theta \).

Step 1: Replace with reciprocal identities

Since \( \csc \theta \) can be written as \( \frac{1}{\sin \theta} \) and \( \tan \theta \) can be written as \( \frac{\sin \theta}{\cos \theta} \), the left side of our equation can be rewritten as \( \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} \).

Step 2: Simplify

By simplifying \( \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} \), we get \( \frac{1}{\cos \theta} \).

Step 3: Replace with reciprocal identity

Finally, the expression \( \frac{1}{\cos \theta} \) is the reciprocal identity for \( \sec \theta \). Therefore, \( \csc \theta \tan \theta \) transforms into \( \sec \theta \).

Key Concepts

Reciprocal Identities

Understanding reciprocal identities is crucial to mastering trigonometry. These identities show the relationship between basic trigonometric functions and their reciprocals. For instance, the reciprocal identities include \( \sin(\theta) = \frac{1}{\csc(\theta)} \) and \( \cos(\theta) = \frac{1}{\sec(\theta)} \), among others. Think of them as flips: if you flip a fraction, you have its reciprocal. It’s the same idea with these trigonometric functions. These identities are incredibly useful for transforming complex expressions into simpler ones, as you can flip between different forms based on what you need for your calculation. When an expression contains a \( \csc(\theta) \), \( \sec(\theta) \), or \( \cot(\theta) \), consider flipping them to their basic forms to make the expression more manageable.

The exercise demonstrated the use of a reciprocal identity by converting \( \csc(\theta) \) into \( \frac{1}{\sin(\theta)} \). This pivotal move set the stage for simplifying the entire equation and is an example of why reciprocal identities are a staple in a trigonometric toolkit.

Simplifying Trigonometric Expressions

Simplification in trigonometry can oftentimes feel like clearing a tangled web. To navigate through this, focus on operations you already know from algebra and apply them to trigonometric terms. Multiplication, division, and factoring are your friends here.

For instance, when faced with something like \( \frac{1}{\sin \theta} \cdot \frac{\sin \theta}{\cos \theta} \), see the opportunity for simplification. The \( \sin \theta \) in the numerator and denominator perfectly cancel each other out due to being reciprocal terms. It becomes key to recognize these opportunities, leading to \( \frac{1}{\cos \theta} \). This simplified form, achieved in the provided solution, reveals a more direct path to showcasing the original equation's truth. Remember, the goal of simplification is to peel back the layers of a trigonometric expression to reveal its more straightforward and fundamental form. Keep an eye out for such reciprocal pairs; they are your simplification allies!

Trigonometry Problem Solving

Successfully tackling trigonometry problems requires a blend of memorization, pattern recognition, and strategic thinking. A solid grasp of trigonometric identities — which act like keys to unlock complex equations — forms the basis of this problem-solving skill. The process often involves looking at one side of an equation and manipulating it to resemble the other side. Strategies include:

• Identifying structures within the problem that point towards specific identities or simplifications.
• Replacing complex expressions with their equivalent simpler identities.
• Strategically using algebraic operations to condense or expand expressions.
• Double-checking that each step maintains the integrity of the equation, especially when applying operations.

In the end, the core principle is to achieve the simplest form that leads to an evident match between both sides of the equation. For example, in the provided exercise, we transformed \( \csc \theta \tan \theta \) into \( \sec \theta \) by first implementing reciprocal identities, then simplifying the expression, and finally, substituting back with a known identity. This illustrates the methodical approach toward problem solving in trigonometry: a step-by-step unraveling of complexities to showcase the underlying direct relationships.