Problem 48 - Basic Technical Mathematics with Calculus

Question

Explain why the given statements are true for an acute angle \(\theta\).The value of \(\sec \theta\) is never less than 1.

Step-by-Step Solution

Answer

For acute angles, \( \sec \theta = \frac{1}{\cos \theta} \) and since \( \cos \theta \leq 1 \), \( \sec \theta \geq 1 \).

Step 1: Define Secant Function

The secant function is defined as the reciprocal of the cosine function. Therefore, \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Consider Values of Cosine for Acute Angles

For an acute angle \( \theta \), the values of \( \cos \theta \) lie between 0 and 1 (exclusive 0). This is because an angle is acute when it is between 0 and 90 degrees.

Step 3: Calculate Minimum Secant Value

Since \( \cos \theta \) is a positive number less than or equal to 1 for an acute angle, \( \sec \theta = \frac{1}{\cos \theta} \) must be greater than or equal to 1. The smallest value of \( \sec \theta \) occurs when \( \cos \theta = 1 \), giving \( \sec \theta = 1 \).

Step 4: Verify with Cosine-Range Dependence

As \( \cos \theta \) decreases (approaches 0 but remains within (0,1) for acute angles), \( \sec \theta \) increases, as the value of \( \sec \theta = \frac{1}{\cos \theta} \) becomes larger. Thus, \( \sec \theta \) cannot be less than 1.

Key Concepts

Understanding Acute Angles

Acute angles are perhaps the easiest angles to visualize. These are angles that measure greater than 0 degrees but less than 90 degrees. In mathematical terms, an acute angle is always more than 0° and less than 90°, or in radians, more than 0 and less than \( \frac{\pi}{2} \). They are found in various geometric shapes, such as triangles, especially in right triangles where the two non-right angles must be acute.
Why are acute angles important in trigonometry? Because they help us understand the behavior of trigonometric functions wherever these functions are defined.

• All angles in a triangle sum up to 180°.
• At least two angles in any given triangle must be acute.

Exploring the Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is directly linked to the cosine function, as it is defined as the reciprocal of cosine. Mathematically, it's expressed as \( \sec \theta = \frac{1}{\cos \theta} \).
This function plays an essential role in trigonometry, particularly in understanding angles and lengths in circles and triangles.

• Secant is undefined when cosine is zero because division by zero is undefined.
• As cosine approaches zero, secant tends to infinity.

For acute angles, since \( \cos \theta \) is always positive and less than or equal to one, \( \sec \theta \) will always be at least 1 and never less.

Unpacking the Cosine Function

To grasp the secant function, you must first understand its foundation: the cosine function. The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse.

• For an acute angle, \( \cos \theta \) yields values between 0 and 1.
• This means that the angle is not a right angle and less than the maximum of 90°.

The significance of the cosine function extends from basic triangle problems to more complex analysis in wave functions and oscillations. It aids in calculating lengths and angles in triangles, contributing greatly to the foundational skills in trigonometry and geometry.

Clarifying Angle Measurement

Angle measurement is a crucial aspect of geometry and trigonometry. Angles can be measured in degrees or radians, which are the two most common units used.

• Degrees are more familiar and useful in everyday contexts, where a full circle is divided into 360 degrees.
• Radians offer a more natural mathematical measure, where a circle is divided into \(2\pi\) radians.

Understanding angle measurement is essential for applying trigonometric functions accurately, including sine, cosine, and secant. This allows for precise calculations in more advanced topics, like calculus and physics, providing a generalized method for measuring rotations and angles.