Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Use the Pythagorean identities to write the expression as an integer. (a) \(\csc ^{2} 3 \alpha-\cot ^{2} 3 \alpha\) (b) \(3 \csc ^{2} \alpha-3 \cot ^{2} \alpha\)

Step-by-Step Solution

Answer

(a) is 1 and (b) is 3.

Step 1: Understand the Pythagorean Identity

Recall the Pythagorean identity for cosecant and cotangent: \[ \csc^2 x - \cot^2 x = 1 \]This identity will allow us to simplify the given expressions as integers.

Step 2: Simplify Expression (a)

Apply the identity to expression \( \csc^2 3\alpha - \cot^2 3\alpha \).Using the identity, we have:\[ \csc^2 3\alpha - \cot^2 3\alpha = 1 \].

Step 3: Simplify Expression (b)

Examine expression \( 3\csc^2 \alpha - 3\cot^2 \alpha \).Factor out the common factor of 3:\[ 3(\csc^2 \alpha - \cot^2 \alpha) \].Using the identity:\[ \csc^2 \alpha - \cot^2 \alpha = 1 \],multiply by 3:\[ 3 \times 1 = 3 \].

Key Concepts

Understanding Cosecant

Cosecant is a trigonometric function that's important to understand. It is the reciprocal of the sine function. For any angle \( \theta \), cosecant is expressed as \( \csc \theta = \frac{1}{\sin \theta} \). Since the sine function gives the ratio of the opposite side to the hypotenuse in a right triangle, cosecant will provide the ratio of the hypotenuse to the opposite side.
This means cosecant values are always greater than or equal to 1, because the hypotenuse is always the longest side in a triangle. It's crucial to remember this when solving problems because it's closely linked with other identities like the Pythagorean identities, which we used in the exercise above. Keeping these relationships in mind can make it much easier to simplify trigonometric expressions.

Exploring Cotangent

Cotangent is another trigonometric function you need to know about. It is the reciprocal of the tangent function. For any angle \( \theta \), cotangent is expressed as \( \cot \theta = \frac{1}{\tan \theta} \) or \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
Cotangent represents the ratio of the adjacent side to the opposite side in a right triangle. Like cosecant, it's helpful in simplifications because it's interrelated with other trigonometric identities. Recall that tangent is the ratio of sine to cosine. This fundamental relationship aids in making the connection to Pythagorean identities used in the step-by-step solution.
Using these identities correctly allows us to work with functions like cosecant and cotangent easily, as demonstrated in the exercise where we applied the identity \( \csc^2 x - \cot^2 x = 1 \) to simplify complex expressions.

The Art of Trigonometric Simplification

Trigonometric simplification involves converting complex trigonometric expressions into simpler ones using trigonometric identities. The aim is often to reduce an expression to a basic trigonometric function value or even an integer. The Pythagorean identities play a crucial role here.
For instance, knowing that \( \csc^2 x - \cot^2 x = 1 \) allows us to simplify expressions like \( \csc^2 3\alpha - \cot^2 3\alpha \) directly to 1. This saves time and effort during calculations.

• Identify the identity that fits your expression.
• Apply simplification by substituting the trigonometric identity into your expression.
• Use basic algebraic steps to finalize the simplification, as we did by factoring out 3 in the second problem \( 3\csc^2 \alpha - 3\cot^2 \alpha \).

By recognizing patterns and reoccurring forms in trigonometry, you will be able to simplify many expressions seamlessly, making problem solving much more efficient and manageable.