Evaluate The Six Trigonometric Functions For Each Value Of

Evaluating trigonometric functions for specific values is a fundamental concept in trigonometry, bridging the gap between abstract definitions and concrete numerical results. Mastering this skill unlocks a deeper understanding of periodic phenomena, oscillations, and wave behavior, all of which are prevalent in fields like physics, engineering, and computer science. This article provides a comprehensive guide to evaluating the six trigonometric functions for various values, covering essential definitions, special angles, techniques for general angles, and practical applications.

Understanding the Six Trigonometric Functions

The six trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)—relate the angles of a right triangle to the ratios of its sides. They are defined as follows:

Sine (sin θ): The ratio of the length of the opposite side to the length of the hypotenuse. sin θ = Opposite / Hypotenuse
Cosine (cos θ): The ratio of the length of the adjacent side to the length of the hypotenuse. cos θ = Adjacent / Hypotenuse
Tangent (tan θ): The ratio of the length of the opposite side to the length of the adjacent side. tan θ = Opposite / Adjacent = sin θ / cos θ
Cosecant (csc θ): The reciprocal of the sine function. csc θ = Hypotenuse / Opposite = 1 / sin θ
Secant (sec θ): The reciprocal of the cosine function. sec θ = Hypotenuse / Adjacent = 1 / cos θ
Cotangent (cot θ): The reciprocal of the tangent function. cot θ = Adjacent / Opposite = 1 / tan θ = cos θ / sin θ

These definitions are based on a right triangle. However, the trigonometric functions can be extended to any angle using the unit circle.

The Unit Circle and Trigonometric Functions

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. An angle θ is formed by rotating a ray from the positive x-axis counterclockwise. The point where this ray intersects the unit circle has coordinates (x, y). In this context:

cos θ = x
sin θ = y

From these definitions, all other trigonometric functions can be derived:

tan θ = y/x
csc θ = 1/y
sec θ = 1/x
cot θ = x/y

The unit circle provides a visual and intuitive way to understand how trigonometric functions behave for different angles, including those beyond the range of a right triangle (0° to 90° or 0 to π/2 radians).

Evaluating Trigonometric Functions for Special Angles

Certain angles, known as special angles, occur frequently in trigonometry and have exact trigonometric values that are useful to memorize or derive. These angles are typically 0°, 30°, 45°, 60°, and 90° (or 0, π/6, π/4, π/3, and π/2 radians).

Here's a table summarizing the trigonometric values for these special angles:

Angle (θ)	sin θ	cos θ	tan θ	csc θ	sec θ	cot θ
0° (0 rad)	0	1	0	undefined	1	undefined
30° (π/6 rad)	1/2	√3/2	√3/3	2	2√3/3	√3
45° (π/4 rad)	√2/2	√2/2	1	√2	√2	1
60° (π/3 rad)	√3/2	1/2	√3	2√3/3	2	√3/3
90° (π/2 rad)	1	0	undefined	1	undefined	0

Deriving the Values:

0° and 90°: Consider the unit circle. At 0°, the point is (1, 0), so cos 0° = 1 and sin 0° = 0. At 90°, the point is (0, 1), so cos 90° = 0 and sin 90° = 1. The other values follow from the definitions. Note that tangent and cotangent are undefined where cosine or sine, respectively, are zero.
45°: A 45-45-90 triangle is an isosceles right triangle. If the legs have length 1, the hypotenuse has length √2. Thus, sin 45° = cos 45° = 1/√2 = √2/2, and tan 45° = 1.
30° and 60°: These angles are found in a 30-60-90 triangle. If the shorter leg (opposite the 30° angle) has length 1, the hypotenuse has length 2, and the longer leg (opposite the 60° angle) has length √3. Thus, sin 30° = 1/2, cos 30° = √3/2, sin 60° = √3/2, and cos 60° = 1/2.

Understanding how to derive these values is crucial, as it reinforces the fundamental relationships between the angles and side ratios.

Evaluating Trigonometric Functions for General Angles

For angles that are not special angles, we can use several techniques to evaluate the trigonometric functions:

1. Reference Angles:

The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. Using reference angles simplifies the evaluation process because trigonometric functions of the reference angle have the same absolute value as the trigonometric functions of the original angle. The only difference might be the sign, which is determined by the quadrant in which the terminal side of the original angle lies.

Steps to find the reference angle and evaluate trigonometric functions:

Determine the Quadrant: Identify which quadrant the angle θ lies in.
Calculate the Reference Angle (α):
- Quadrant I: α = θ
- Quadrant II: α = 180° - θ (or π - θ in radians)
- Quadrant III: α = θ - 180° (or θ - π in radians)
- Quadrant IV: α = 360° - θ (or 2π - θ in radians)
Evaluate the Trigonometric Function of the Reference Angle: Find the value of the trigonometric function for the reference angle α.
Determine the Sign: Based on the quadrant of the original angle θ, determine the correct sign for the trigonometric function. Use the mnemonic "ASTC" (All Students Take Calculus):
- Quadrant I: All trigonometric functions are positive.
- Quadrant II: Sine (and cosecant) are positive.
- Quadrant III: Tangent (and cotangent) are positive.
- Quadrant IV: Cosine (and secant) are positive.
Apply the Sign: Apply the appropriate sign to the value obtained in step 3.

Example: Evaluate sin(210°)

Quadrant: 210° lies in Quadrant III.
Reference Angle: α = 210° - 180° = 30°
Evaluate: sin(30°) = 1/2
Sign: In Quadrant III, sine is negative.
Apply Sign: sin(210°) = -1/2

2. Using Trigonometric Identities:

Trigonometric identities are equations that are true for all values of the variables for which the expressions are defined. These identities can be used to simplify expressions and to find the values of trigonometric functions. Some key identities include:

Pythagorean Identities:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Angle Sum and Difference Identities:
- sin(A + B) = sinA cosB + cosA sinB
- sin(A - B) = sinA cosB - cosA sinB
- cos(A + B) = cosA cosB - sinA sinB
- cos(A - B) = cosA cosB + sinA sinB
- tan(A + B) = (tanA + tanB) / (1 - tanA tanB)
- tan(A - B) = (tanA - tanB) / (1 + tanA tanB)
Double-Angle Identities:
- sin(2θ) = 2 sinθ cosθ
- cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
- tan(2θ) = (2 tanθ) / (1 - tan²θ)
Half-Angle Identities:
- sin(θ/2) = ±√((1 - cosθ) / 2)
- cos(θ/2) = ±√((1 + cosθ) / 2)
- tan(θ/2) = ±√((1 - cosθ) / (1 + cosθ)) = sinθ / (1 + cosθ) = (1 - cosθ) / sinθ

Example: Find sin(15°) using the difference identity.

sin(15°) = sin(45° - 30°)
sin(45° - 30°) = sin(45°)cos(30°) - cos(45°)sin(30°)
sin(45° - 30°) = (√2/2)(√3/2) - (√2/2)(1/2)
sin(15°) = (√6 - √2) / 4

3. Calculators and Software:

For angles that do not lend themselves to easy calculation using reference angles or identities, calculators or software can be used to approximate the values of trigonometric functions. Ensure the calculator is in the correct mode (degrees or radians) before performing the calculation.

Quadrantal Angles

Quadrantal angles are angles that are integer multiples of 90° (π/2 radians). These angles lie on the axes of the coordinate plane: 0°, 90°, 180°, 270°, and 360° (0, π/2, π, 3π/2, and 2π radians).

The trigonometric values for these angles are:

Angle (θ)	sin θ	cos θ	tan θ	csc θ	sec θ	cot θ
0° (0 rad)	0	1	0	undefined	1	undefined
90° (π/2 rad)	1	0	undefined	1	undefined	0
180° (π rad)	0	-1	0	undefined	-1	undefined
270° (3π/2 rad)	-1	0	undefined	-1	undefined	0
360° (2π rad)	0	1	0	undefined	1	undefined

Notice that at these angles, either sine or cosine is 0, leading to undefined values for tangent, cotangent, secant, and cosecant.

Even and Odd Trigonometric Functions

Understanding the properties of even and odd functions can simplify calculations:

Even Function: A function f(x) is even if f(-x) = f(x). Cosine and secant are even functions:
- cos(-θ) = cos(θ)
- sec(-θ) = sec(θ)
Odd Function: A function f(x) is odd if f(-x) = -f(x). Sine, tangent, cosecant, and cotangent are odd functions:
- sin(-θ) = -sin(θ)
- tan(-θ) = -tan(θ)
- csc(-θ) = -csc(θ)
- cot(-θ) = -cot(θ)

These properties allow us to easily evaluate trigonometric functions for negative angles.

Example: Evaluate sin(-30°)

Since sine is an odd function, sin(-30°) = -sin(30°) = -1/2

Practical Applications

Evaluating trigonometric functions is not just a mathematical exercise; it has wide-ranging applications in various fields:

Physics: Analyzing wave motion (sound waves, light waves), projectile motion, and simple harmonic motion. Trigonometric functions are essential for describing oscillations and periodic phenomena.
Engineering: Designing structures, analyzing circuits, and processing signals. For example, in electrical engineering, sinusoidal functions are used to model alternating current (AC).
Navigation: Calculating distances and bearings using triangulation and spherical trigonometry. GPS systems rely heavily on trigonometric calculations.
Computer Graphics: Creating realistic images and animations. Trigonometric functions are used to rotate, scale, and translate objects in 3D space.
Music: Understanding the mathematical relationships between musical notes and harmonies.
Astronomy: Modeling planetary orbits and celestial phenomena.

Common Mistakes to Avoid

Incorrect Quadrant: Failing to correctly identify the quadrant of the angle, leading to incorrect signs for the trigonometric functions.
Radian vs. Degree Mode: Using the incorrect mode (degrees or radians) on a calculator.
Undefined Values: Forgetting that tangent, cotangent, secant, and cosecant are undefined at certain angles.
Incorrect Identities: Misapplying trigonometric identities.
Memorization vs. Understanding: Relying solely on memorization without understanding the underlying concepts. Focus on understanding the unit circle, reference angles, and the relationships between the trigonometric functions.

Examples and Practice Problems

Example 1: Evaluate all six trigonometric functions for θ = 5π/6.

Quadrant: 5π/6 is in Quadrant II.
Reference Angle: α = π - 5π/6 = π/6
sin(5π/6) = sin(π/6) = 1/2 (Sine is positive in Quadrant II)
cos(5π/6) = -cos(π/6) = -√3/2 (Cosine is negative in Quadrant II)
tan(5π/6) = sin(5π/6) / cos(5π/6) = (1/2) / (-√3/2) = -√3/3
csc(5π/6) = 1 / sin(5π/6) = 2
sec(5π/6) = 1 / cos(5π/6) = -2√3/3
cot(5π/6) = 1 / tan(5π/6) = -√3

Example 2: Given cos θ = -3/5 and θ is in Quadrant III, find 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 Quadrant III, sin θ is negative. Therefore, sin θ = -4/5
tan θ = sin θ / cos θ = (-4/5) / (-3/5) = 4/3

Practice Problems:

Evaluate sin(3π/4), cos(7π/6), and tan(5π/3).
Find all six trigonometric functions for θ = -π/3.
If sin θ = 12/13 and θ is in Quadrant I, find cos θ and tan θ.
If tan θ = -5/12 and θ is in Quadrant II, find sin θ and cos θ.
Evaluate sin(105°) using the sum identity.

Conclusion

Evaluating trigonometric functions for specific values is a fundamental skill in mathematics and its applications. By understanding the definitions of the six trigonometric functions, the unit circle, special angles, reference angles, trigonometric identities, and the properties of even and odd functions, you can effectively evaluate trigonometric functions for a wide range of angles. Remember to pay attention to the quadrant of the angle to determine the correct sign and to avoid common mistakes. Consistent practice is key to mastering this skill and unlocking a deeper understanding of trigonometry and its applications in various scientific and engineering disciplines.