Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\cot 25^{\circ} \cdot \csc 65^{\circ} \cdot \sin 25^{\circ}$$

Short answer

The exact value is 1.

Step-by-step solution

- Break Down the Expression

The given expression is \( \cot 25^{\circ} \cdot \csc 65^{\circ} \cdot \sin 25^{\circ} \). Ensure to recognize each trigonometric function.

- Apply Complementary Angle Theorem

Recall that complementary angles satisfy the identity: \( \sin(\90^{\circ} - \theta) = \cos(\theta) \). Thus, \( \csc 65^{\circ} = \csc(\90^{\circ} - 25^{\circ}) = \sec 25^{\circ} \).

- Substitute Complementary Identities

Rewrite the expression using the identity found: \( \cot 25^{\circ} \cdot \sec 25^{\circ} \cdot \sin 25^{\circ} \).

- Apply Fundamental Identities

Using fundamental trigonometric identities: \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \). So, substitute to get \( \frac{\cos 25^{\circ}}{\sin 25^{\circ}} \cdot \frac{1}{\cos 25^{\circ}} \cdot \sin 25^{\circ} \).

- Simplify the Expression

Simplify the terms by canceling out common factors: \( \frac{\cos 25^{\circ}}{\sin 25^{\circ}} \cdot \frac{1}{\cos 25^{\circ}} \cdot \sin 25^{\circ} = \frac{1}{\sin 25^{\circ}} \cdot \sin 25^{\circ} = 1 \).

Key concepts

Complementary Angle Theorem

The Complementary Angle Theorem is an essential concept in trigonometry that says for any angle \( \theta \), \( \theta \) and its complementary angle \( 90^{\circ} - \theta \) have related trigonometric functions. This theorem can be very helpful for simplifying and solving trigonometric expressions.

For instance, if you know the value of \( \sin(\theta) \), you can find \( \cos(90^{\circ}-\theta) \) because they are equal. The same goes for other trigonometric functions like cosecant (\( \csc \)) and secant (\( \sec \)).

This was used in the original solution to transform \( \csc 65^{\circ} \) into \( \sec 25^{\circ} \). Recognizing these relationships makes solving trigonometric problems more manageable and often reduces the complexity of the calculations necessary.

Fundamental Identities

Fundamental Identities are the building blocks of trigonometry. They include basic relationships between the trigonometric functions that always hold true. Some of the most important ones are:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Quotient Identities: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)
• Reciprocal Identities: \( \csc(\theta) = \frac{1}{\sin(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \)

These identities help simplify the expressions and are crucial for solving the given exercise. By applying these identities, we transformed the original complex expression into manageable parts:

\( \cot(25^{\circ}) \) can be rewritten using the quotient identity: \( \frac{\cos(25^{\circ})}{\sin(25^{\circ})} \).
Similarly, \( \sec(25^{\circ}) \) was rewritten using the reciprocal identity: \( \frac{1}{\cos(25^{\circ})} \).
These transformations ultimately made it easier to simplify the expression and find the solution.

Trigonometric Functions

Trigonometric functions describe the relationships between the angles and the lengths of sides in right-angled triangles. The most common trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).

Here’s a quick rundown:

• \(\text{\sin}\theta = \text{opposite}/\text{hypotenuse}\)
• \(\text{\cos}\theta = \text{adjacent}/\text{hypotenuse}\)
• \(\text{\tan}\theta = \text{opposite}/\text{adjacent}\)
• \(\text{\csc}\theta = 1/\text{\sin}\theta\)
• \(\text{\sec}\theta = 1/\text{\cos}\theta\)
• \(\text{\cot}\theta = 1/\text{\tan}\theta\)

In the original exercise, these definitions and relationships allowed us to break down the trigonometric expressions and solve them using substitution. For example, knowing that \( \cot(25^{\circ}) = \frac{\cos(25^{\circ})}{\sin(25^{\circ})} \) and \( \sec(25^{\circ}) = \frac{1}{\cos(25^{\circ})} \) permitted us to simplify the expression step by step.

Understanding these basic trigonometric functions and how they relate to each other is crucial for simplifying complex trigonometric expressions and solving exercises accurately.