Problem 51 - Precalculus Essentials

Question

Express the exact value of each function as a single fraction. Do not use a calculator. $$\text { If } \theta \text { is an acute angle and } \cot \theta=\frac{1}{4}, \text { find } \tan \left(\frac{\pi}{2}-\theta\right)$$.

Step-by-Step Solution

Answer

Therefore, the exact value of \( \tan \left(\frac{\pi}{2} - \theta\right) \) is \( \frac{1}{\sqrt{17}} \).

Step 1: Understanding the Problem

The problem is asking to find the tangent of an angle that is related to the given angle \( \theta \). Whenever the angles are related by \( \frac{\pi}{2} \) it suggests that the co-function identity might be useful. In this case, given \( \cot \theta = \frac{1}{4} \) and to find \( \tan(\frac{\pi}{2} - \theta) \). Remember, cotangent and tangent are reciprocals, so \( \tan \theta = 4 \).

Step 2: Applying Co-function Identity

Subtract the angle \( \theta \) from \( \frac{\pi}{2} \). As \( \tan \left(\frac{\pi}{2}- \theta\right) \) is equivalent to \( \cos \theta \). Hence, we replace \( \tan \theta = 4 \) with \( \cos \theta \). As the relationship between cosine and secant is also reciprocal or \( \cos \theta = \frac{1}{\sec \theta} \). We can replace \( \cos \theta \) with \( \frac{1}{\sec \theta} \).

Step 3: Applying Pythagorean Identity

The Pythagorean identity of a trigonometric function for secant \( \sec \theta \) is \( \sec \theta = \sqrt{1 + \tan^2 \theta} \). We know \( \tan \theta = 4 \). Substituting the value of \( \tan \theta \) into the Pythagorean identity equation, we get \( \sec \theta = \sqrt{1 + 4^2} = \sqrt{17} \). Sub this value back into Step 2 to find \( \cos \theta = \frac{1}{\sqrt{17}} \).

Key Concepts

Understanding Acute Angles

In trigonometry, an acute angle is one that is less than 90 degrees or less than \( \frac{\pi}{2} \) radians.
This is important because the function values for acute angles are always positive.
When dealing with trigonometric identities and equations, knowing whether an angle is acute can often simplify problems.

• Acute angles are crucial in simplifying trigonometric functions as they always lie within the first quadrant.
• Within the first quadrant, the values of sine, cosine, tangent, and their reciprocals are always positive.

The problem starts by stating that \( \theta \) is an acute angle, ensuring that all function values will be positive.
This is helpful when applying identities like co-function identities where sign matters.

Co-Function Identities Explained

Co-function identities relate the trigonometric functions of complementary angles.
Essentially, for two angles \( A \) and \( B \) that add up to \( \frac{\pi}{2} \), the co-function identity helps switch between functions.

• Example: \( \tan \left( \frac{\pi}{2} - \theta \right) = \cot \theta \)
• This identity utilizes the complement relationship, aiming to simplify expressions involving angles.
• Key co-function pairs are: sine-cosine and tangent-cotangent.

In this exercise, the identity \( \tan \left( \frac{\pi}{2} - \theta \right) = \cot \theta \) allows us to directly use \( \cot \theta \) when finding \( \tan \left( \frac{\pi}{2} - \theta \right) \).
This is particularly helpful because it translates the problem into one that uses the given cotangent value, simplifying the task.

Pythagorean Identity in Trigonometry

The Pythagorean identity is an essential trigonometric identity derived from the Pythagorean theorem.
It connects the squares of sine, cosine, and tangent, particularly for angles in the unit circle.

• Primary identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• For tangent and secant: \( 1 + \tan^2 \theta = \sec^2 \theta \)

In this exercise, knowing that \( \tan \theta = 4 \), we can plug this into the identity \( \sec^2 \theta = 1 + \tan^2 \theta \).
Solving gives us \( \sec \theta = \sqrt{17} \), which leads to finding \( \cos \theta = \frac{1}{\sec \theta} = \frac{1}{\sqrt{17}} \).
This identity not only simplifies the expression but also helps to find exact values of trigonometric functions without a calculator.