Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\sec \theta=2\) (b) \(\cot \theta=1\)

Step-by-Step Solution

Answer

The answers are: (a) \(\theta = 60^\circ\) or \(\frac{\pi}{3}\) radians, (b) \(\theta = 45^\circ\) or \(\frac{\pi}{4}\) radians.

Step 1: Solve for \(\theta\) for \(\sec \theta = 2\)

The secant function is the reciprocal of the cosine function. So, for \(\sec \theta = 2\), it implies that \(\cos \theta = \frac{1}{2}\). The cosine of what angle in the first quadrant gives the result as \(\frac{1}{2}\)? The answer is \(60^\circ\) or \(\frac{\pi}{3}\) radians.

Step 2: Solve for \(\theta\) for \(\cot \theta = 1\)

The cotangent is the reciprocal of the tangent function. So, for \(\cot \theta = 1\), it implies that \(\tan \theta = 1\). The tangent of what angle in the first quadrant gives the result \(1\)? The answer is \(45^\circ\) or \(\frac{\pi}{4}\) radians.

Key Concepts

Secant Function

The secant function, written as \(\sec \theta\), is a trigonometric function. It's the reciprocal of the cosine function. This means that \(\sec \theta = \frac{1}{\cos \theta}\). When you solve for \(\theta\) in a problem where \(\sec \theta = 2\), you're essentially looking for a value of \(\theta\) such that \(\cos \theta = \frac{1}{2}\).
This approach helps you find the angle \(\theta\) by determining which angle in the trigonometric circle has a cosine of \(\frac{1}{2}\).
The angles we often deal with are in the unit circle, and for \(\cos \theta = \frac{1}{2}\), the angle \(\theta\) is \(60^\circ\) or \(\frac{\pi}{3}\) radians.

• \(\sec \theta\) is undefined when \(\cos \theta = 0\) because division by zero is undefined.
• The secant function is useful in various fields, including physics, engineering, and geometry.

Cotangent Function

The cotangent function, denoted as \(\cot \theta\), is another trigonometric ratio. It's the reciprocal of the tangent function: \(\cot \theta = \frac{1}{\tan \theta}\).
When a problem states that \(\cot \theta = 1\), it implies that \(\tan \theta = 1\). So, you need to identify the angle where the tangent is 1 in the first quadrant.
This angle is \(45^\circ\) or \(\frac{\pi}{4}\) radians. In this special case, both sine and cosine yield equal values, which results in a tangent of 1.
It's important to remember:

• \(\cot \theta\) becomes undefined when \(\tan \theta = 0\), as division by zero is not defined.
• The cotangent function often appears in the analysis of periodic phenomena and oscillations.

Radians and Degrees

Radians and degrees are two units for measuring angles. Each has a specific use and importance:
- **Degrees** are more common in everyday scenarios. A complete circle is 360 degrees.
- **Radians** are often used in mathematics and physics due to their natural relationship with circular motion and periodic functions. One full circle is \(2\pi\) radians.

To convert between degrees and radians, you can use the relationship:

• Radians to Degrees: Multiply by \(\frac{180}{\pi}\)
• Degrees to Radians: Multiply by \(\frac{\pi}{180}\)

Understanding this conversion helps when working with problems involving trigonometric functions, as different disciplines might prefer different units.
For instance, an angle of \(60^\circ\) converts to \(\frac{\pi}{3}\) radians, and \(45^\circ\) converts to \(\frac{\pi}{4}\) radians.