Problem 70 - 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) \(\csc \theta=\frac{2 \sqrt{3}}{3}\) (b) \(\sin \theta=\frac{1}{2}\)

Step-by-Step Solution

Answer

(a) \(\theta= 60^{\circ}\) or \(\frac{\pi}{3}\) radians; (b) \(\theta= 30^{\circ}\) or \(\frac{\pi}{6}\) radians

Step 1: Problem (a): Resolve \(\csc \theta\)

Cosecant (\(\csc\)) is the reciprocal of Sine (\(\sin\)). So, if \(\csc \theta=\frac{2 \sqrt{3}}{3}\), then \(\sin \theta= \frac{3}{2\sqrt{3}}\). This can be simplified as \(\sin \theta= \frac{\sqrt{3}}{2}\). We know this value corresponds to \(\theta= 60^{\circ}\) or \(\frac{\pi}{3}\) radians.

Step 2: Problem (b): Resolve \(\sin \theta\)

Here, \(\sin \theta= \frac{1}{2}\) is a direct value from the trigonometric table. Its corresponding value is \(\theta= 30^{\circ}\) or \(\frac{\pi}{6}\) radians.

Step 3: Consider the quadrant

Since the given boundaries are first quadrant (0 to 90 degrees or 0 to \(\frac{\pi}{2}\) radians), the obtained angles for both problems (a) and (b) are valid without any change because these angles are already in the first quadrant.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is one of the six primary trigonometric functions. It is the reciprocal of the sine function. When you have a sine value, you can find its cosecant by using the formula:

• \( \csc \theta = \frac{1}{\sin \theta} \).

This means that if you know the sine of an angle, flipping or inverting that value will give you the cosecant. In trigonometry, it is particularly useful when dealing with right triangles, where the cosecant function helps in determining the ratio of the hypotenuse to the opposite side of the angle \( \theta \). In problem (a), we found \( \csc \theta=\frac{2 \sqrt{3}}{3} \). By flipping this value, we derived that \( \sin \theta = \frac{\sqrt{3}}{2} \). This simplification is often necessary for solving trigonometric equations. Knowing how to work with reciprocals can greatly expand your understanding and ability to solve problems involving trigonometric functions.

Sine Function

The sine function, represented as \( \sin \theta \), is foundational in trigonometry and is defined as the ratio of the length of the opposite side of an angle in a right triangle to the hypotenuse. Essentially, it helps in identifying the vertical component of an angle’s projection.For a more practical sense, the sine function describes how the y-coordinate of a point on a unit circle changes as \( \theta \) varies. The function values are often found using known angles or through them being part of established trigonometric tables.In the given exercise, problem (b) provided \( \sin \theta = \frac{1}{2} \). This is a key value that corresponds to a 30° angle or \( \frac{\pi}{6} \) radians. Memorizing such common sine values is beneficial as these are frequently encountered in calculations. Applying this understanding can aid one in deducing unknown angles quickly, without extensive calculations.

Degrees and Radians

Degrees and radians are two units used to measure angles. They are crucial for understanding different scenarios where angles play a role, such as trigonometry or geometry.

• **Degrees** are commonly used in many everyday applications and are measured in parts of a complete circle, which is 360°. Therefore, a right angle is 90°.
• **Radians** pivot around the concept of angle subtended at the center of a circle by an arc whose length is equal to the circle's radius. One complete revolution around a circle is \( 2\pi \) radians. Thus, 180° corresponds to \( \pi \) radians.

Switching between these units is crucial for solving problems efficiently, especially when working with mathematical formulas that are simplified when using radians. In stepping from degrees to radians and back, remember the conversion:

• \(1\text{ radian} = \frac{180}{\pi} \text{ degrees} \),
• \(1\text{ degree} = \frac{\pi}{180} \text{ radians} \).

In our exercise, knowing that both angle solutions — 30° and 60° — have equivalent radian measures \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \) respectively, confirms our understanding of their relationship and application across different contexts.