Find The Exact Value Of The Trigonometric Function.

Finding the Exact Value of Trigonometric Functions: A Comprehensive Guide

Determining the exact value of trigonometric functions is a fundamental skill in mathematics, particularly crucial for calculus, trigonometry, and other advanced mathematical fields. While calculators provide approximate values, understanding how to find exact values is essential for developing a deep comprehension of trigonometric relationships and solving complex problems. This comprehensive guide delves into various methods and techniques to accurately determine the exact values of trigonometric functions for specific angles.

Understanding the Unit Circle

The unit circle is an indispensable tool for determining the exact values of trigonometric functions. This circle, with a radius of 1 unit, is centered at the origin (0,0) of a coordinate plane. Each point on the unit circle can be represented by its coordinates (cos θ, sin θ), where θ is the angle formed between the positive x-axis and the line segment connecting the origin to that point.

Key Angles and Their Coordinates

Several key angles on the unit circle have easily calculable coordinates, forming the basis for determining exact trigonometric values. These include:

• 0° (0 radians): (1, 0) Therefore, cos 0° = 1 and sin 0° = 0.
• 30° (π/6 radians): (√3/2, 1/2)
• 45° (π/4 radians): (√2/2, √2/2)
• 60° (π/3 radians): (1/2, √3/2)
• 90° (π/2 radians): (0, 1)
• 120° (2π/3 radians): (-1/2, √3/2)
• 135° (3π/4 radians): (-√2/2, √2/2)
• 150° (5π/6 radians): (-√3/2, 1/2)
• 180° (π radians): (-1, 0)
• 270° (3π/2 radians): (0, -1)
• 360° (2π radians): (1, 0)

Memorizing these coordinates is crucial for efficiently solving problems involving exact trigonometric values. Understanding the symmetry of the unit circle also aids in determining values for angles beyond these key values.

Utilizing Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved angles. These identities provide alternative ways to express trigonometric functions, enabling us to find exact values indirectly.

Fundamental Identities

Some fundamental identities are:

• sin²θ + cos²θ = 1: This Pythagorean identity is exceptionally useful for finding the value of one trigonometric function if the value of the other is known.
• tan θ = sin θ / cos θ: The tangent function is the ratio of the sine and cosine functions.
• cot θ = cos θ / sin θ: The cotangent is the reciprocal of the tangent.
• sec θ = 1 / cos θ: The secant is the reciprocal of the cosine.
• csc θ = 1 / sin θ: The cosecant is the reciprocal of the sine.

By manipulating these identities, we can often simplify complex expressions and find the exact values more easily.

Angle Sum and Difference Identities

These identities are powerful tools for determining the exact values of trigonometric functions for angles that are sums or differences of the key angles:

• sin(A ± B) = sin A cos B ± cos 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)

By substituting the known values of the key angles (A and B), we can determine the exact values of the trigonometric functions for the resulting angles.

Double and Half-Angle Identities

These identities are useful for finding the exact values for angles that are double or half the key angles:

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ - sin²θ = 1 - 2sin²θ = 2cos²θ - 1
• tan 2θ = 2 tan θ / (1 - tan²θ)
• 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

The choice of plus or minus sign in the half-angle formulas depends on the quadrant in which θ/2 lies.

Examples of Finding Exact Trigonometric Values

Let's work through some examples to illustrate the application of these concepts:

Example 1: Find the exact value of sin 75°.

75° can be expressed as 45° + 30°. Using the angle sum identity:

sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4

Example 2: Find the exact value of cos 15°.

15° can be expressed as (30°/2). Using the half-angle identity:

cos 15° = cos (30°/2) = ±√[(1 + cos 30°)/2] = ±√[(1 + √3/2)/2] = √(2 + √3) / 2 (We choose the positive sign because 15° is in the first quadrant).

Example 3: Find the exact value of tan (5π/12).

5π/12 radians is equivalent to 75°. We can use the angle sum identity or the fact that 5π/12 = π/4 + π/6 and proceed similarly to Example 1.

Example 4: Given that cos θ = -3/5 and θ is in the third quadrant, find the exact value of sin θ and tan θ.

Using the Pythagorean identity, sin²θ + cos²θ = 1:

sin²θ + (-3/5)² = 1 => sin²θ = 1 - 9/25 = 16/25 => sin θ = ±4/5

Since θ is in the third quadrant, sin θ = -4/5.

Then, tan θ = sin θ / cos θ = (-4/5) / (-3/5) = 4/3

Advanced Techniques and Considerations

For angles beyond the easily derived values, we might need to employ more advanced techniques, such as:

• Using reference angles: The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. The trigonometric functions of the original angle are related to the trigonometric functions of the reference angle through their signs (positive or negative), depending on the quadrant.
• Product-to-sum and sum-to-product identities: These identities help transform expressions involving products or sums of trigonometric functions into more manageable forms.
• Complex numbers: The use of Euler's formula (e^(iθ) = cos θ + i sin θ) connects trigonometric functions with complex exponentials, often simplifying complex calculations.

Conclusion

Finding the exact values of trigonometric functions requires a solid understanding of the unit circle, trigonometric identities, and a systematic approach to problem-solving. Mastering these techniques is crucial for success in higher-level mathematics and related fields. Remember to utilize the key angles, apply relevant identities strategically, and be mindful of the quadrant in which the angle lies to determine the correct sign of the trigonometric function. Through consistent practice and application, you can develop proficiency in finding exact trigonometric values, significantly enhancing your mathematical skills and problem-solving abilities. Continuous practice and revisiting these methods will reinforce your understanding and build confidence in tackling even more complex trigonometric problems.