Problem 61 - 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) \(\csc \theta=\frac{2 \sqrt{3}}{3}\) (b) \(\sin \theta=\frac{\sqrt{2}}{2}\)

Step-by-Step Solution

Answer

(a) \( \theta = 60^{\circ} \) or \( \frac{\pi}{3} \) radian. (b) \( \theta = 45^{\circ} \) or \( \frac{\pi}{4} \) radian.

Step 1: Analyzing given for \(\csc \theta\)

Given \(\csc \theta=\frac{2 \sqrt{3}}{3}\). The cosecant is the reciprocal of the sine function. Hence, \(\sin \theta = \frac{3}{2 \sqrt{3}}\). Rationalizing the denominator by multiplying by \(\sqrt{3}\) produces \(\sin \theta = \frac{\sqrt{3}}{2}\).

Step 2: Computing \(\theta\) for given \(\sin \theta\)

Considering the defined range of \( \theta \) [\(0^{\circ}<\theta<90^{\circ}\) or \(0<\theta<\pi / 2\)], the only angle for which the sine value is \(\frac{\sqrt{3}}{2}\) is \(\theta=60^{\circ}\) or \(\frac{\pi}{3}\) radian.

Step 3: Analyzing given for \(\sin \theta\)

Given \(\sin \theta=\frac{\sqrt{2}}{2}\).

Step 4: Computing \(\theta\) for given \(\sin \theta\)

Considering the defined range of \( \theta \) [\(0^{\circ}<\theta<90^{\circ}\) or \(0<\theta<\pi / 2\)], the only angle for which the sine value is \(\frac{\sqrt{2}}{2}\) is \(\theta=45^{\circ}\) or \(\frac{\pi}{4}\) radian.

Key Concepts

Cosecant

The cosecant function, often abbreviated as "csc," is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. This means if you have \(\sin \theta = x\), then \(\csc \theta = \frac{1}{x}\). Cosecant is not as commonly used as sine, cosine, or tangent, but it is still important. In our exercise, we have \(\csc \theta = \frac{2 \sqrt{3}}{3}\).

• To solve for \( \theta \), convert cosecant to sine by taking the reciprocal: \(\sin \theta = \frac{3}{2 \sqrt{3}}\).

Understanding the relationship between cosecant and sine is key to solving trigonometric problems that involve these functions. It allows us to find the corresponding angle \( \theta \).

Sine function

The sine function is fundamental in trigonometry and is represented by the sine ratio, \(\sin \theta\). For any angle \( \theta \), it describes the ratio of the length of the opposite side to the hypotenuse in a right triangle.

• In the exercise, we convert \(\csc \theta = \frac{2 \sqrt{3}}{3}\) to \(\sin \theta = \frac{\sqrt{3}}{2}\).
• When given \(\sin \theta = \frac{\sqrt{2}}{2}\), it translates into knowing which angle provides this ratio.

Recognizing these common values and their corresponding angles, such as \(60^{\circ}\) or \(\frac{\pi}{3}\) and \(45^{\circ}\) or \(\frac{\pi}{4}\), is essential for solving trigonometric equations effortlessly.

Angle conversion

Angle conversion involves changing an angle measurement from one unit to another. The most common units are degrees and radians.

• Degrees are based on dividing a circle into 360 equal parts.
• Radians relate a circle's radius to its arc length and there are \(2\pi\) radians in a circle.

To convert from degrees to radians, multiply the degree measurement by \(\frac{\pi}{180}\). Conversely, convert radians to degrees by multiplying the radian value by \(\frac{180}{\pi}\). In this exercise:

• \(60^{\circ}\) converts to \(\frac{\pi}{3}\) radians.
• \(45^{\circ}\) converts to \(\frac{\pi}{4}\) radians.

Understanding these conversions is crucial for moving between different problem settings in trigonometry.

Rationalizing the denominator

Rationalizing the denominator is a process used to make a fraction more simple, particularly when it contains a radical (square root) in the denominator. Here’s how it works:

• Multiply both the numerator and the denominator by the needed radical term to remove it from the denominator.

For example, if you have \(\sin \theta = \frac{3}{2 \sqrt{3}}\), you rationalize by multiplying the entire fraction by \(\sqrt{3}\):

• This gives you \(\sin \theta = \frac{3 \sqrt{3}}{2 \times 3} = \frac{\sqrt{3}}{2}\).

Rationalizing the denominator makes calculations easier and helps achieve more practical representations in mathematical expressions. This technique is frequently used in trigonometry to simplify expressions involving trigonometric ratios.