Problem 11 - Precalculus

Question

Use the definitions (not a calculator) to evaluate the six trigonometric functions of each angle. If a value is undefined, state this. $$-2 \pi$$

Step-by-Step Solution

Answer

Sine: 0, Cosine: 1, Tangent: 0, Cosecant: undefined, Secant: 1, Cotangent: undefined.

Step 1: Understanding the Angle

The angle is \(-2\pi\), which represents a full circle rotation in the negative (clockwise) direction. Since one full rotation is \(2\pi\), this angle coincides with \(0\) radians on the unit circle.

Step 2: Determine Coordinates on the Unit Circle

At \(0\) radians, the coordinates on the unit circle are \((1, 0)\). This is crucial as we'll use these coordinates to find all six trigonometric functions.

Step 3: Calculate Sine Function

Using the definition of sine, \(\sin(\theta) = y\), we have \(\sin(-2\pi) = 0\).

Step 4: Calculate Cosine Function

Using the definition of cosine, \(\cos(\theta) = x\), we have \(\cos(-2\pi) = 1\).

Step 5: Calculate Tangent Function

Using the definition of tangent, \(\tan(\theta) = \frac{y}{x}\), we find \(\tan(-2\pi) = \frac{0}{1} = 0\).

Step 6: Calculate Cosecant Function

The cosecant is the reciprocal of sine, \(\csc(\theta) = \frac{1}{\sin(\theta)}\). Because \(\sin(-2\pi) = 0\), \(\csc(-2\pi)\) is undefined.

Step 7: Calculate Secant Function

The secant is the reciprocal of cosine, \(\sec(\theta) = \frac{1}{\cos(\theta)}\). Therefore, \(\sec(-2\pi) = \frac{1}{1} = 1\).

Step 8: Calculate Cotangent Function

The cotangent is the reciprocal of tangent, \(\cot(\theta) = \frac{1}{\tan(\theta)}\). Since \(\tan(-2\pi) = 0\), \(\cot(-2\pi)\) is undefined.

Key Concepts

Unit Circle

The unit circle is a foundational concept in trigonometry. It is a circle with a radius of one unit centered at the origin of a coordinate plane. The unit circle allows us to understand the behavior of trigonometric functions by correlating angles and coordinates.

When discussing angles on the unit circle, we often start from the positive x-axis and move counterclockwise for positive angles. Clockwise movement means negative angles. Each angle corresponds to a unique point on the unit circle, with coordinates \(x, y\).

• For example, at \(0\) radians, the point is \(1, 0\).
• At \(\frac{\pi}{2}\) radians, the coordinates are \(0, 1\).
• At \(\pi\) radians, the coordinates are \(-1, 0\).
• At \(\frac{3\pi}{2}\), the coordinates become \(0, -1\).

Understanding where an angle lands on the unit circle can help derive the sine and cosine functions and subsequently help evaluate angles without a calculator.

Radians

Radians are a measure of angles based on the radius of the circle. Unlike degrees, which divide a circle into 360 equal parts, radians divide it into \(2\pi\) parts. This is because the circumference of a unit circle is \(2\pi\) units.

One full circle rotation is \(2\pi\) radians, similar to \(360\) degrees. Radians are the standard unit of angular measure in mathematics because they provide a more natural relationship between the length of a circle's radius and the arc.

• An angle of \(-2\pi\) radians means you move one full revolution in the clockwise direction.
• This places you back at the starting position, equivalent to \(0\) radians.
• Using radians helps us easily find circle-related properties, such as the arc length, by multiplying the angle in radians by the radius.

Radians provide a powerful way to work with periodic functions and are integral to understanding many complex mathematical and physical phenomena.

Reciprocal Trigonometric Functions

In trigonometry, reciprocal functions are essential and complement the primary trigonometric functions. They demonstrate the inverses of sine, cosine, and tangent, and include cosecant, secant, and cotangent functions.

The cosecant function (\(\csc\)) is the reciprocal of sine. If \(\sin(\theta) = \text{value}\), then \(\csc(\theta) = \frac{1}{\text{value}}\). However, if sine is zero, such as at zero radians, cosecant becomes undefined because you can't divide by zero.

• The secant function (\(\sec\theta\)) is the reciprocal of cosine. \(\sec(\theta) = \frac{1}{\cos(\theta)}\). Since \(\cos(0) = 1\), \(\sec(0)\) is also 1.
• The cotangent function (\(\cot\theta\)) is the reciprocal of tangent. \(\cot(\theta) = \frac{1}{\tan(\theta)}\). Like cosecant, cotangent is undefined when the tangent is zero.

Understanding reciprocal functions allows us to expand our mathematical toolbox when solving trigonometric equations, especially when encountering undefined values and seeking the most efficient solution.