Problem 24 - Algebra 2 and Trigonometry

Question

In \(24-32,\) find the exact value of each expression. $$ \sin (\arctan 1) $$

Step-by-Step Solution

Answer

\( \sin(\arctan(1)) = \frac{\sqrt{2}}{2} \).

Step 1: Understand the Inverse Function

The expression involves the function \( \arctan(1) \). The \( \arctan \) function returns the angle whose tangent is 1. Recall that \( \tan(\frac{\pi}{4}) = 1 \). Thus, \( \arctan(1) = \frac{\pi}{4} \).

Step 2: Evaluate the Sine Function

Now evaluate \( \sin(\arctan(1)) \), which becomes \( \sin\left(\frac{\pi}{4}\right) \) based on the outcome from step 1. We need to find the sine of the angle \( \frac{\pi}{4} \).

Step 3: Calculate \(\sin\left(\frac{\pi}{4}\right)\)

The angle \( \frac{\pi}{4} \) radians (or 45 degrees) has a sine value of \( \frac{\sqrt{2}}{2} \). This is a standard value for the sine function at this angle.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are mathematical functions that reverse trigonometric functions like sine, cosine, and tangent. When you see a function like \( \arctan(x) \), it's asking for the angle whose tangent value equals \( x \). In our featured exercise, we utilized \( \arctan(1) \). Let's break it down further:

• \( \arctan(1) \) determines the angle whose tangent is 1.
• From trigonometry, we know that \( \tan(\pi/4) = 1 \), which simplifies \( \arctan(1) \) to \( \pi/4 \).

By transforming a trigonometric value back to an angle, inverse trigonometric functions provide a powerful tool for solving various problems.

Sine Function

The sine function, often denoted as \( \sin \), is one of the primary trigonometric functions. It takes an angle and gives a numerical value that represents the ratio of the opposite side to the hypotenuse in a right triangle. In the current exercise, after finding the angle using \( \arctan(1) \), we move to evaluate \( \sin(\pi/4) \).

• When \( \sin \) is applied to the angle \( \pi/4 \) (which is 45 degrees), it evaluates to \( \sin(\pi/4) = \frac{\sqrt{2}}{2} \).
• This specific value is a part of the standard unit circle values and is often memorized due to its frequent occurrence.

Understanding this relationship helps solve problems where angle values are needed after calculating with inverse trigonometric results.

Radian Measure

Radians are a way to measure angles based on the radius of a circle. They offer a natural measure for angles, especially in trigonometry. Here's why radians are key in the given exercise:

• We worked with angles in radians \( \pi/4 \), which equates to 45 degrees.
• The unit circle is typically expressed in radians, making concepts easier to visualize and compute.
• One full rotation around a circle is \( 2\pi \) radians, equivalent to 360 degrees.

Grasping radians is crucial for understanding how trigonometric functions behave as they use radians to map angles efficiently within the field of mathematics.