Problem 62 - Precalculus: A Concise Course

Question

Find the values of \(\theta\) in degrees \(\left(0^{\circ}<\theta<90^{\circ}\right)\) and radians \((0<\theta<\pi / 2)\) without the aid of a calculator. (a) \(\cot \theta=\frac{\sqrt{3}}{3}\) (b) \(\sec \theta=\sqrt{2}\)

Step-by-Step Solution

Answer

For part (a), where \(\cot \theta=\frac{\sqrt{3}}{3}\), \(\theta\) is equal to \(60^{\circ}\) or \(\frac{\pi}{3}\) radians. For part (b), where \(\sec \theta=\sqrt{2}\), \(\theta\) is \(45^{\circ}\) or \(\frac{\pi}{4}\) radians.

Step 1: Identifying the Trigonometric Ratios for Part (a)

The cotangent of an angle is the reciprocal of the tangent of that angle, meaning \(\cot \theta=\frac{1}{\tan \theta}\). We're given that \(\cot \theta=\frac{\sqrt{3}}{3}\). Therefore, \(\tan \theta=\frac{1}{\cot \theta}=3/\sqrt{3}= \sqrt{3}\). The angle whose tangent is \(\sqrt{3}\) is \(\theta=60^{\circ}\). Convert this to radians by multiplying by \(\frac{\pi}{180}\), to get \(\theta=\frac{\pi}{3}\) rad.

Step 2: Identifying the Trigonometric Ratios for Part (b)

The secant of an angle is the reciprocal of the cosine of that angle, i.e., \(\sec \theta=\frac{1}{\cos \theta}\). We're given that \(\sec \theta=\sqrt{2}\), therefore, \(\cos \theta=\frac{1}{\sec \theta}=1/\sqrt{2}=\frac{\sqrt{2}}{2}\). Using the basic cosine values, the angle is found to be \(\theta=45^{\circ}\). Convert this to radians by multiplying by \(\frac{\pi}{180}\), giving \(\theta=\frac{\pi}{4}\) rad.

Step 3: Consolidating Findings

The solutions to the exercise have been found. In (a), for \(\cot \theta=\frac{\sqrt{3}}{3}\), the value of \(\theta\) is \(60^{\circ}\) or \(\frac{\pi}{3}\) rad. In (b), for \(\sec \theta=\sqrt{2}\), the value of \(\theta\) is \(45^{\circ}\) or \(\frac{\pi}{4}\) rad.

Key Concepts

Cotangent

In trigonometry, the cotangent function is an essential concept that helps relate the angles of a triangle to the ratios of its sides. Cotangent is the reciprocal of the tangent. The formula for cotangent is given by \[ \cot \theta = \frac{1}{\tan \theta} \]This means, if you have the tangent of an angle, you can easily find the cotangent by taking the reciprocal. For instance, if \( \cot \theta = \frac{\sqrt{3}}{3} \), to find \( \tan \theta \), just flip that fraction, leading to \( \tan \theta = \sqrt{3} \).

• Cotangent is particularly useful in specific angle calculations without a calculator.
• Understanding reciprocal relationships helps in proving identities or simplifying trigonometric expressions.

In our case, given the tangent value, we identify the angle \( \theta \), which yields \( \theta = 60^{\circ} \). We arrive at this by knowing typical angle tangents such as \( \tan 60^{\circ} = \sqrt{3} \). This method is essential for solving trigonometric equations efficiently.

Secant

Secant is another reciprocal trigonometric function that plays a vital role in trigonometry. Secant is the reciprocal of the cosine function and is defined as \[ \sec \theta = \frac{1}{\cos \theta} \]Knowing one allows you to solve for the other easily. In the exercise, you are given \( \sec \theta = \sqrt{2} \). Taking the reciprocal provides \( \cos \theta = \frac{1}{\sqrt{2}} \), which simplifies to \( \cos \theta = \frac{\sqrt{2}}{2} \).

• Having a deep understanding of secant, especially around key angles, simplifies many problems.
• Secant often relates to scenarios where alternative cosine forms are needed.

In the provided problem, the cosine value \( \cos \theta = \frac{\sqrt{2}}{2} \) aligns with the angle \( \theta = 45^{\circ} \). Leveraging known angle values concerning cosine and their reciprocals can be very beneficial in problem-solving.

Angle Conversion

Angle conversion is an important skill in trigonometry and mathematics in general. It combines geometry and trigonometry, allowing users to transition between degree measures and radian measures comfortably. To convert degrees to radians, use the conversion factor \[ \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]Here are steps to follow:

• Multiply the angle in degrees by \( \frac{\pi}{180} \).
• Simplify the resulting fraction if possible.
• Express the angle in terms of \( \pi \).

In the exercise, the conversion of \( 60^{\circ} \) gives \( \frac{\pi}{3} \) and \( 45^{\circ} \) converts to \( \frac{\pi}{4} \). Regular practice with these conversions solidifies understanding and boosts confidence in exchanging between angle measurement systems.

Degrees to Radians

Understanding degrees and radians is foundational in trigonometry. Degrees measure angle sizes using a circle divided into 360 parts, while radians measure angle sizes as parts of a circle's circumference based on the radius. Importantly, \[ 360^{\circ} = 2\pi \text{ radians} \]This relationship derives the conversion factor frequently used in trigonometry calculations. Converting degrees to radians involves the formula \[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]Practicing this conversion through examples strengthens understanding.

• An angle of \( 60^{\circ} \) becomes \( \frac{\pi}{3} \) radians.
• An angle of \( 45^{\circ} \) becomes \( \frac{\pi}{4} \) radians.

Learning to interchange angles between these two systems is crucial in various physics and engineering applications, providing a deeper understanding of movement and periodic functions.