Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-14,\) find each value of \(\theta : \mathbf{a} .\) in degrees \(\mathbf{b} .\) in radians $$ \theta=\arcsin 1 $$

Step-by-Step Solution

Answer

\(\theta = 90^\circ\) or \(\theta = \frac{\pi}{2}\) radians.

Step 1: Understanding the Problem

We are asked to find the value of \(\theta\) for the expression \(\theta = \arcsin 1\). This means we need to find the angle \(\theta\) for which the sine is equal to 1.

Step 2: Finding the Angle in Degrees

The function \(\arcsin x\) (or inverse sine) returns the angle in the range of \(-90^\circ \leq \theta \leq 90^\circ\) where the sine of the angle \(\theta\) is equal to \(x\). Since \(\sin 90^\circ = 1\), \(\arcsin 1 = 90^\circ\). Therefore, \(\theta = 90^\circ\).

Step 3: Converting Degrees to Radians

To convert from degrees to radians, we use the conversion factor \(\frac{\pi}{180}\). So, \(\theta = 90^\circ \times \frac{\pi}{180} = \frac{\pi}{2}\). This means \(\arcsin 1 = \frac{\pi}{2}\) radians.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools for solving trigonometric equations. Specifically, they allow us to determine angles when the values of their trigonometric functions are known. For instance, the arcsine function, denoted as \( \arcsin x \), helps us find the angle \( \theta \) for which \( \sin \theta = x \). This function is particularly useful in situations where the sine of an angle is known, and we need to determine the actual angle itself.

• The range of \( \arcsin x \) is between \(-90^\circ\) and \(90^\circ\) or \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) in radians. This range ensures that for each sine value, there is a unique corresponding angle.
• Understanding inverse trigonometric functions is crucial for further studies in calculus, physics, and engineering, where precise angle measurements are necessary.

When using \( \arcsin 1 \), we find that the angle \( \theta \) is \(90^\circ\) because \( \sin 90^\circ = 1 \).

Radians and Degrees Conversion

Converting between radians and degrees is a vital skill in trigonometry. Degrees and radians are both units for measuring angles, with radians being the standard unit in higher mathematics.

• To convert from degrees to radians, multiply by \( \frac{\pi}{180} \). This factor stems from the equivalence of \(180^\circ\) to \(\pi\) radians.
• Conversely, to convert from radians to degrees, multiply by \( \frac{180}{\pi} \).

Let's look at an example: to convert \(90^\circ\) to radians, we calculate \( 90^\circ \times \frac{\pi}{180} = \frac{\pi}{2} \). This conversion shows how degrees can be expressed in terms of radians, which is useful for trigonometric calculations in mathematics and physics.

Trigonometric Equations

Trigonometric equations involve finding angles or other quantities that satisfy a trigonometric relationship. Solving these equations often requires a solid understanding of the properties of trigonometric functions.

• These equations frequently crop up in physics, engineering, and other sciences, where wave patterns and cycles of motion are described.
• For example, the equation \( \theta = \arcsin 1 \) is simple but illustrates the principle: here, we find \( \theta = 90^\circ \) or \( \theta = \frac{\pi}{2} \), as these are the angles where the sine reaches 1 within the defined range of the \( \arcsin \) function.

By solving trigonometric equations, you can determine critical angle values in various scientific and engineering contexts. Each solution gives a unique insight into the harmony between geometric shapes and their trigonometric properties.