Finding The Exact Value Of Cos(25°) / Sin(65°) A Step-by-Step Guide

July 6, 2025 by Scholario Team 68 views

Find the Exact Value of the Expression: A Comprehensive Guide

Finding the exact value of trigonometric expressions without using a calculator can seem daunting at first. However, by understanding fundamental trigonometric identities, complementary angle relationships, and special angle values, we can simplify complex expressions and arrive at precise solutions. This comprehensive guide will walk you through the process of evaluating the expression $\frac{\cos(25^{\circ})}{\sin(65^{\circ})}$ step-by-step, while also providing a broader understanding of the underlying concepts and techniques.

Understanding the Basics of Trigonometry

At the heart of trigonometry lies the relationship between angles and the sides of right-angled triangles. The three primary trigonometric functions—sine (sin), cosine (cos), and tangent (tan)—define these relationships. For an acute angle θ in a right-angled triangle:

Sine (sin θ) is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
Cosine (cos θ) is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan θ) is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

These definitions form the bedrock of trigonometric calculations. Furthermore, several key identities and relationships build upon these definitions, allowing us to manipulate and simplify trigonometric expressions. One such crucial concept is the complementary angle relationship.

Complementary Angle Relationships

Complementary angles are two angles whose sum is 90 degrees. A fundamental property of trigonometric functions relates the sine and cosine of complementary angles. Specifically, the sine of an angle is equal to the cosine of its complement, and vice versa. Mathematically, this can be expressed as:

sin(θ) = cos(90° - θ)
cos(θ) = sin(90° - θ)

This relationship is crucial for simplifying expressions and solving trigonometric equations. It allows us to convert between sine and cosine functions, which is particularly useful when dealing with angles that are not standard values (such as 30°, 45°, and 60°). In the given problem, we can leverage this relationship to simplify the expression $\frac{\cos(25^{\circ})}{\sin(65^{\circ})}$ .

Step-by-Step Solution for $\frac{\cos(25^{\circ})}{\sin(65^{\circ})}$

Now, let's apply the complementary angle relationship to solve the given expression. Our goal is to simplify the expression $\frac{\cos(25^{\circ})}{\sin(65^{\circ})}$ without using a calculator. The key lies in recognizing that 25° and 65° are complementary angles because their sum is 90° (25° + 65° = 90°).

Identify Complementary Angles: Recognize that 25° and 65° are complementary angles.
Apply the Complementary Angle Relationship: Use the identity sin(θ) = cos(90° - θ) to rewrite the denominator. We can express sin(65°) as cos(90° - 65°), which simplifies to cos(25°).

So, sin(65°) = cos(90° - 65°) = cos(25°).
Substitute and Simplify: Substitute the equivalent expression back into the original expression:

$\frac{\cos(25^{\circ})}{\sin(65^{\circ})} = \frac{\cos(25^{\circ})}{\cos(25^{\circ})}$
Evaluate: Since the numerator and the denominator are now the same, the expression simplifies to 1.

$\frac{\cos(25^{\circ})}{\cos(25^{\circ})} = 1$

Therefore, the exact value of the expression $\frac{\cos(25^{\circ})}{\sin(65^{\circ})}$ is 1. This solution highlights the power of understanding and applying trigonometric identities.

Expanding Your Trigonometric Toolkit

While the complementary angle relationship was sufficient to solve the given problem, several other trigonometric identities and techniques can be invaluable in tackling more complex expressions. Some of these include:

Pythagorean Identities

The Pythagorean identities are derived from the Pythagorean theorem and relate the squares of trigonometric functions. The most fundamental Pythagorean identity is:

sin²(θ) + cos²(θ) = 1

From this identity, we can derive two other useful identities:

1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)

These identities are crucial for simplifying expressions involving squares of trigonometric functions and can often be used to rewrite expressions in a more manageable form.

Sum and Difference Formulas

The sum and difference formulas allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. These formulas are particularly useful when dealing with angles that can be expressed as sums or differences of special angles (such as 30°, 45°, and 60°). The key formulas are:

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Double-Angle and Half-Angle Formulas

Double-angle and half-angle formulas are special cases of the sum and difference formulas and provide expressions for trigonometric functions of 2θ and θ/2 in terms of trigonometric functions of θ. These formulas are essential for simplifying expressions involving multiple angles or fractions of angles. The key formulas are:

sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
tan(2θ) = (2tan(θ)) / (1 - tan²(θ))
sin(θ/2) = ±√((1 - cos(θ)) / 2)
cos(θ/2) = ±√((1 + cos(θ)) / 2)
tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ))) = sin(θ) / (1 + cos(θ)) = (1 - cos(θ)) / sin(θ)

Tips for Mastering Trigonometric Simplification

To effectively simplify trigonometric expressions without a calculator, consider the following tips:

Memorize Key Identities: Familiarize yourself with the fundamental trigonometric identities, including reciprocal identities, quotient identities, Pythagorean identities, sum and difference formulas, and double-angle and half-angle formulas. Regular practice and memorization are crucial for quick recall.
Recognize Complementary Angles: Be vigilant in identifying complementary angles, as the relationship between their trigonometric functions often provides a direct simplification path.
Convert to Sine and Cosine: When faced with complex expressions involving multiple trigonometric functions, consider converting all functions to sine and cosine. This often simplifies the expression and reveals opportunities for applying other identities.
Look for Squares: Pay attention to squared trigonometric functions, as they often suggest the use of Pythagorean identities. Substituting and simplifying using Pythagorean identities can significantly reduce the complexity of the expression.
Practice Regularly: Consistent practice is the key to mastering trigonometric simplification. Work through a variety of problems, gradually increasing the complexity, to build your skills and confidence.

Conclusion: The Power of Trigonometric Identities

Finding the exact value of the expression $\frac{\cos(25^{\circ})}{\sin(65^{\circ})}$ without a calculator demonstrates the power and elegance of trigonometric identities. By understanding and applying the complementary angle relationship, we were able to simplify the expression and arrive at the solution, which is 1. This problem serves as a valuable illustration of how trigonometric identities can be used to manipulate and simplify complex expressions, making them more manageable and solvable. Mastery of these techniques not only enables you to solve specific problems but also provides a deeper understanding of the interconnectedness of trigonometric concepts. As you continue to explore trigonometry, remember that practice and a solid grasp of fundamental identities are the keys to success.

By mastering these identities and practicing regularly, you'll be well-equipped to tackle a wide range of trigonometric simplification problems. Trigonometry is not just about memorizing formulas; it's about understanding the relationships between angles and sides, and using that understanding to solve problems creatively and efficiently.