Question

Use the unit circle to evaluate the six trigonometric functions of \(\theta\). \(\theta=\frac{7 \pi}{2}\)

Short answer

\(\sin(\theta) = -1\), \(\cos(\theta) = 0\), \(\tan(\theta)\) = undefined, \(\csec(\theta) = -1\), \(\sec(\theta)\) = undefined, \(\cot(\theta) = 0\)

Step-by-step solution

Identify the Equivalent Angle

Given that our trigonometric functions repeat every \(2\pi\) (full circle), an angle of \(\frac{7 \pi}{2}\) is equivalent to an angle of \(\frac{3 \pi}{2}\) as it would end up at the same point on the unit circle.

Locate the Equivalent Angle on Unit Circle

Locate \(\frac{3 \pi}{2}\) on the unit circle. This angle results in a point at the negative y-axis of the unit circle, which coordinates are (0,-1).

Evaluate Trigonometric Functions

Use the obtained point (0,-1) to evaluate the six trigonometric functions. Since the x-coordinate is 0 and the y-coordinate is -1, we have: \ - \(\sin(\theta)\) = -1 \ - \(\cos(\theta)\) = 0 \ - \(\tan(\theta)\) = \(\frac{\sin(\theta)}{\cos(\theta)}\) = undefined, as we cannot divide by zero. \ - \(\csc(\theta)\) = \(\frac{1}{\sin(\theta)}\) = -1 \ - \(\sec(\theta)\) = \(\frac{1}{\cos(\theta)}\) = undefined, as we cannot divide by zero. \ - \(\cot(\theta)\) = \(\frac{\cos(\theta)}{\sin(\theta)}\) = 0, since 0 divided by any number except zero itself is equal to zero.

Key concepts

Understanding the Unit Circle

The unit circle is a crucial concept in trigonometry, representing all the angles and their corresponding values on a circle with a radius of one. When you plot an angle, the circle's center is at the origin \(0,0\) on a coordinate plane. Each point on the unit circle can be described by its coordinates \(x, y\), which correspond to the cosine and sine values of that angle.

Key characteristics of the unit circle:

• The circumference is \(2\pi\), representing a full rotation.
• The radius is always 1.
• Coordinates on the circle represent cosine (x-value) and sine (y-value).

For instance, an angle like \(\frac{3\pi}{2}\) is found on the negative y-axis since it's 270° counterclockwise from the positive x-axis. These points help us determine various trigonometric function values.

Angle Conversion Explained

Angle conversion is the practice of expressing an angle in different units, primarily radians or degrees. Converting between these units helps in understanding angles in different contexts. One full circle is equivalent to 360 degrees or \(2\pi\) radians.

For conversions, remember that:

• rac{ ext{degrees}}{360} = rac{ ext{radians}}{2\pi}
• 1 radian = approximately 57.3 degrees.

To convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\). Conversely, multiply the degree measure by \(\frac{\pi}{180}\) to convert it to radians. As an example, \(\frac{7\pi}{2}\) is easier to interpret by converting it and recognizing that we've completed multiple full rotations.

Understanding Sine and Cosine

Sine and cosine are foundational trigonometric functions that arise from the unit circle. They describe the vertical (y-coordinate) and horizontal (x-coordinate) distances of a point on the circle, respectively.

Let's break them down:

• Sine ( ext{sin}): This function gives the y-coordinate of a point on the unit circle. It's essential in determining how far up or down the point is from the center.
• Cosine ( ext{cos}): Provides the x-coordinate, indicating how far to the left or right the point is from the center.

For the angle \(\frac{3\pi}{2}\), sine is -1 as it is the y-coordinate, and cosine is 0 as it is the x-coordinate. Understanding these functions helps in graphing waves and solving complex trigonometric equations.

Tangent and Cotangent Functions

Tangent and cotangent functions are derived from the sine and cosine functions. They reveal the slope or steepness of the line connecting the point on the circle to the origin.

Tangent ( ext{tan}) is the ratio of sine to cosine. It reflects how much a line rises or falls relative to its run:

• an(\theta) = rac{ ext{sin}(\theta)}{ ext{cos}(\theta)}

Cotangent ( ext{cot}) is the reciprocal of tangent, offering a complementary perspective:

• ext{cot}(\theta) = rac{ ext{cos}(\theta)}{ ext{sin}(\theta)}

These calculations can be tricky when cosine equals zero, such as \(\frac{3\pi}{2}\), making tangent undefined and cotangent zero.

Cosecant and Secant Functions

Cosecant and secant are reciprocal functions of sine and cosine, providing insight into inversions of these familiar functions.

Cosecant ( ext{csc}) and secant ( ext{sec}) are defined as:

• ext{csc}(\theta) = rac{1}{ ext{sin}(\theta)}
• ext{sec}(\theta) = rac{1}{ ext{cos}(\theta)}

These inverses are particularly useful when sine or cosine is zero, which would otherwise be undefined in their reciprocal form. At \(\frac{3\pi}{2}\), cosecant is -1 because sine is -1, while secant is undefined since cosine is zero. These relationships are powerful tools, especially when simplifying expressions or solving trigonometric equations.