Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the value of the expression without using a calculator. $$\sin \left[\cos ^{-1}(-1)+\pi\right]$$

Step-by-Step Solution

Answer

The value of the expression \(\sin [\cos^{-1}(-1)+\pi]\) is 0.

Step 1: Find the angle using inverse cosine

Recall that the unit circle gives the cosine of an angle as the x-coordinate where the terminal side of the angle intersects the circle. The inverse cosine function, \(cos^{-1}(-1)\), is asking for the angle whose cosine (x-coordinate on the unit circle) equals -1. This angle is \(\pi\) in radians.

Step 2: Add the resultant angle to pi

Now apply the mathematical operation stated in the problem: \(cos^{-1}(-1) + \(\pi\). This gives: \(\pi + \pi = 2\pi\).

Step 3: Evaluate the sine of the resultant angle

Now, calculate the sine of the resultant angle obtained from step 2. The sine of an angle is the y-coordinate where the terminal side of the angle intersects the unit circle. The sine of \(2\pi\) or any multiple of \(2\pi\) is 0, since these angles correspond to points on the unit circle with a y-coordinate of 0. So, \(\sin(2\pi) = 0\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This simple circle, however, holds invaluable information about angles and their trigonometric functions.

Here's how the unit circle connects to trigonometric functions:

• The x-coordinate of any point on the unit circle gives the cosine of the angle \(\theta\).
• The y-coordinate provides the sine of the angle \(\theta\).

By understanding these coordinates, you can find the sine and cosine values of any angle placed on the unit circle.

Angles can be measured in degrees or radians. On the unit circle, a complete rotation around the center corresponds to \(2\pi\) radians. This makes the unit circle a powerful tool for visualizing and understanding trigonometric identities and relationships.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential for solving equations where you need to determine an angle from a given trigonometric value. They reverse the traditional trigonometric functions, such as sine, cosine, and tangent.

For example, the inverse cosine function, denoted as \(\cos^{-1}(x)\), returns the angle whose cosine is \(x\). In other words, if \(\cos(\theta) = x\), then \(\cos^{-1}(x) = \theta\).

When dealing with inverse trigonometric functions, it's important to remember these functions have restricted ranges to return a unique angle:

• \(\cos^{-1}\) has a range of \([0, \pi]\).
• \(\sin^{-1}\) operates within \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• \(\tan^{-1}\) ranges from \((-\frac{\pi}{2}, \frac{\pi}{2})\).

These functions ensure the angles returned are always within specific bounds, providing consistent and reliable results for solving for angles.

Radians

Radians provide a convenient way to measure angles in terms of the arc length of a circle. One radian is the angle formed when the arc length is equal to the radius of the circle.

In terms of conversion, \(\pi\) radians are equivalent to 180 degrees, allowing for easy transition between the two measurement systems:

• \(360\, \text{degrees} = 2\pi \, \text{radians}\).
• \(180\, \text{degrees} = \pi \, \text{radians}\).
• \(90\, \text{degrees} = \frac{\pi}{2} \, \text{radians}\).

Using radians often simplifies mathematical expressions in calculus and higher mathematics as it naturally aligns with the unit circle.

The challenge of switching to radians is mostly initial, as most people first learn angles in degrees. However, recognizing radian measures, especially in trigonometric identities and calculus, greatly enhances understanding and problem-solving flexibility.