Problem 28 - Algebra 2 and Trigonometry

Question

In \(26-33 :\) a. Rewrite each function value in terms of its cofunction. b. Find the exact value of the function value found in a. $$ \tan \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The exact value of \( \tan \frac{\pi}{6} \) is \( \frac{1}{\sqrt{3}} \).

Step 1: Understand Cofunction Identity

The cofunction identity for tangent states that \( \tan(\theta) = \cot(\frac{\pi}{2} - \theta) \). This is key to rewriting the tangent function in terms of its cofunction, which is the cotangent.

Step 2: Apply the Cofunction Identity

Apply the cofunction identity to the given function \( \tan \frac{\pi}{6} \). According to the identity: \[ \tan \frac{\pi}{6} = \cot \left( \frac{\pi}{2} - \frac{\pi}{6} \right) \]

Step 3: Simplify Inside the Function

Simplify the expression inside the cotangent function: \[ \frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]Thus, \[ \tan \frac{\pi}{6} = \cot \frac{\pi}{3} \].

Step 4: Find Exact Value of Cotangent

Now find the exact value of \( \cot \frac{\pi}{3} \). Recall that \( \cot \theta = \frac{1}{\tan \theta} \) and \( \tan \frac{\pi}{3} = \sqrt{3} \). Thus, \[ \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \].

Step 5: Final Step: Conclusion

Therefore, the function value \( \tan \frac{\pi}{6} \) rewritten in terms of its cofunction is \( \cot \frac{\pi}{3} \), and the exact value of this function is \( \frac{1}{\sqrt{3}} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions include sine, cosine, tangent, cotangent, secant, and cosecant. They're incredibly useful, especially when dealing with right-angled triangles. Each function has a specific ratio of sides. For example, the tangent of an angle is the ratio of the opposite side to the adjacent side in a right-angled triangle. A key aspect of understanding trigonometric functions is recognizing how they are interrelated. Cofunction identities, such as the tangent and cotangent relationship, play a crucial role. The world of trigonometry connects beautifully through these identities, making it easier to solve various problems.

Tangent and Cotangent

The tangent function (\( \tan \theta \)) and the cotangent function (\( \cot \theta \)) are both vital in trigonometry. Specifically, they are reciprocal: \( \cot \theta = \frac{1}{\tan \theta} \). This means if you know one, you can find the other easily. Another captivating property is their cofunction identity: \( \tan(\theta) = \cot \left( \frac{\pi}{2} - \theta \right) \), helping us rewrite one in terms of the other, especially when angles are expressed in radians or degrees.
Some items to remember:

• Tangent is undefined at \( \theta = \frac{\pi}{2} \), \( \frac{3\pi}{2} \), and so on, because it involves a division by zero.
• Cotangent is undefined at angles where tangent is zero, such as \( \theta = 0 \), \( \pi \), \( 2\pi \), etc.

Understanding these distinctions and connections makes working with them in trigonometry both intuitive and insightful.

Exact Values in Trigonometry

Finding exact trigonometric values requires some familiarity with the properties of triangles and trigonometry itself. For instance, knowing exact values like \( \tan \frac{\pi}{6} \) and \( \tan \frac{\pi}{3} \) is essential. This knowledge enables the determination of values like \( \cot \frac{\pi}{3} \), using the relationship \( \cot \theta = \frac{1}{\tan \theta} \).
Frequently used exact values include:

• \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \)
• \( \tan \frac{\pi}{3} = \sqrt{3} \)
• \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \)

Remembering these exact values simplifies problem-solving. They act as tools, providing clear and concise results, eliminating the need for calculators in many cases. Understanding these fundamental concepts strengthens your grasp on more advanced trigonometry topics as well.