Problem 35 - Precalculus : Building Concepts and Connections

Question

Find the exact values of all the trigonometric functions for the giocn calues of \(t .\) If a certain value is undefined, state sa Do not use a calculator. $$t=\frac{13 \pi}{3}$$

Step-by-Step Solution

Answer

The exact values of the trigonometric functions for \( t = \frac{13 \pi}{3} \) are \( \sin{t} = \frac{\sqrt{3}}{2} \), \( \cos{t} = \frac{1}{2} \), \( \tan{t} = \sqrt{3} \), \( \csc{t} = \frac{2}{\sqrt{3}} \), \( \sec{t} = 2 \), and \( \cot{t} = \frac{1}{\sqrt{3}} \).

Step 1: Simplify the given value

First note that \( t = \frac{13 \pi}{3} \) can be simplified by noting that \( 2\pi \) is one complete revolution around the unit circle. Therefore, \( t \) can be rewritten by subtracting \( 4\pi \), which is two full circles, to give \( t = \frac{13 \pi}{3} - \frac{12 \pi}{3} = \frac{\pi}{3} \). This is because the responses on the unit circle are periodic with period \( 2\pi \).

Step 2: Calculate the sine and cosine

Knowing that \( t = \frac{\pi}{3} \) corresponds to an angle of 60° in the first quadrant, the exact value for sine and cosine can be retrieved using the unit circle or standard right triangles. Thus, \( \sin{\frac{\pi}{3}} = \frac{\sqrt{3}}{2} \) and \( \cos{\frac{\pi}{3}} = \frac{1}{2} \).

Step 3: Calculate the tangent

The tangent function is the ratio of sine to cosine. Therefore, \( \tan{\frac{\pi}{3}} = \frac{\sin{\frac{\pi}{3}}}{\cos{\frac{\pi}{3}}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \).

Step 4: Compute the cofunctions

The values of the other three functions (cosecant, secant, and cotangent) are the reciprocals of sine, cosine, and tangent, respectively. For \( t = \frac{\pi}{3} \), these are \( \csc{\frac{\pi}{3}} = \frac{2}{\sqrt{3}} \), \( \sec{\frac{\pi}{3}} = 2 \), and \( \cot{\frac{\pi}{3}} = \frac{1}{\sqrt{3}} \), respectively.

Key Concepts

Unit Circle

The unit circle is a vital tool in trigonometry. It aids in understanding the relationship between angles and their corresponding trigonometric values. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. This is the unit circle. Its total circumference is equivalent to the angle made by one complete rotation, which is given by \( 2\pi \) radians.

By using the unit circle, you can simplify trigonometric problems, especially those that do not require a calculator. When you calculate trigonometric functions for angles, you essentially project these points around the circle onto the x or y-axis.

For an angle like \( \frac{\pi}{3} \), found on the unit circle, the x-coordinate gives the cosine value, while the y-coordinate gives the sine value. Hence:

• The cosine of \( \frac{\pi}{3} \) is \( \frac{1}{2} \)
• The sine of \( \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{2} \)

Understanding these concepts allows for the determination that the angle \( \frac{13\pi}{3} \) simplifies to \( \frac{\pi}{3} \) after subtracting multiples of \( 2\pi \). Therefore, the unit circle supports understanding periodicity in trigonometric functions.

Sine and Cosine Identities

Sine and cosine are the foundational trigonometric functions. They express the primary dimensions of a right-angled triangle or the coordinates of an angle on the unit circle. In the context of trigonometric identities, these functions model periodic patterns like waves.

**Key Relationships**:

• The Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Co-function identity: \( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) \) and vice-versa
• Double angle formula: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \)
• Angle sum and difference formulas to solve for non-standard angles

The task of determining sine and cosine for \( \frac{\pi}{3} \) was simplified by recognizing it corresponds to 60 degrees:

• Sine of 60° or \( \frac{\pi}{3} \) = \( \frac{\sqrt{3}}{2} \)
• Cosine of 60° or \( \frac{\pi}{3} \) = \( \frac{1}{2} \)

These identities are crucial for quickly repurposing known angles without needing intensive calculations.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are the flipsides of the basic sine, cosine, and tangent functions. They are: cosecant, secant, and cotangent.

**Definitions**:

• Cosecant (\( \csc \)) is the reciprocal of sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• Secant (\( \sec \)) is the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• Cotangent (\( \cot \)) is the reciprocal of tangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

In the example of \( \frac{\pi}{3} \):

• \( \csc(\frac{\pi}{3}) = \frac{2}{\sqrt{3}} \)
• \( \sec(\frac{\pi}{3}) = 2 \)
• \( \cot(\frac{\pi}{3}) = \frac{1}{\sqrt{3}} \)

These values show how by simply knowing sine, cosine, and tangent, you can easily find their reciprocals. This showcases their interdependence and importance in trigonometry.