Problem 41 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\cos \left(\sin ^{-1} \frac{\sqrt{3}}{2}\right)\)

Step-by-Step Solution

Answer

The exact value is \( \frac{1}{2} \).

Step 1: Understand the Inverse Function

The expression involves finding the cosine of the arcsin (inverse sine). Given \( \sin^{-1} \frac{\sqrt{3}}{2} \), we need to find an angle \( \theta \) such that \( \sin \theta = \frac{\sqrt{3}}{2} \).

Step 2: Identify Theta Using a Right Triangle

The value \( \frac{\sqrt{3}}{2} \) is a known sine value for the angle \( \frac{\pi}{3} \) (or 60 degrees). Therefore, \( \theta = \frac{\pi}{3} \).

Step 3: Calculate Cosine of the Angle

Now, find \( \cos(\theta) \) where \( \theta = \frac{\pi}{3} \). The cosine of \( \frac{\pi}{3} \) is \( \frac{1}{2} \), which can be derived from the unit circle or trigonometric tables.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are special functions that allow us to find angles when given trigonometric ratios. They are the inverses of the standard trigonometric functions: sine, cosine, and tangent. For instance, the inverse sine function, denoted as \( \sin^{-1} \) or arcsin, helps determine an angle whose sine value is known.

There are purposes for using inverse trigonometric functions. They help in:

• Solving triangles when some elements are known.
• Calculating angles in real-world problems, such as inclines and elevations.

When you encounter an inverse trigonometric function, think of it as asking: "What angle produces this specific sine, cosine, or tangent value?"

Always remember that inverse trigonometric functions have restricted ranges. This means they only provide angles within specific intervals. For example, \( \sin^{-1} x \) provides angles between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). Interpret results accordingly in your calculations.

Right Triangle

A right triangle is a triangle with one angle measuring exactly 90 degrees. Understanding right triangles is crucial in trigonometry since many trigonometric principles are based on them. These triangles are defined by three sides:

• Hypotenuse: the longest side, opposite the right angle.
• Opposite: the side opposite the angle of interest.
• Adjacent: the side next to the angle of interest, excluding the hypotenuse.

The basic trigonometric functions often relate angles to their opposite and adjacent sides via the hypotenuse. When working with right triangles and trigonometry, remember:

• Use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
• Solve for missing sides and angles using trig ratios such as \(\sin\), \(\cos\), and \(\tan\).
• Apply these principles to gain insight into angles and dimensions within right triangles.

This concept aids in solving exercises involving angles like the one in the Original Exercise.

Unit Circle

The unit circle is a powerful tool in trigonometry that helps visualize angles and trigonometric functions. It's a circle centered at the origin (0, 0) of the coordinate plane, with a radius of 1.

This simplicity allows easy calculation of sine, cosine, and tangent values based on the coordinates:

• The x-coordinate of a point on the circle is the cosine of the angle.
• The y-coordinate is the sine of the angle.

By utilizing the unit circle, you can find trigonometric values for common angles such as \(0, \frac{\pi}{4}, \frac{\pi}{2}, \) and \(\frac{3\pi}{4}\). This is how you arrive at solutions like knowing that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).

The unit circle also aids in understanding periodicity and symmetry in trig functions. Learning to manipulate the unit circle can greatly enhance your problem-solving skills in trigonometry.

Angle Measurement

Angle measurement is a fundamental concept in trigonometry. It describes how we quantify angles, typically given in degrees or radians. Understanding the conversion between these two units is essential:

• Degrees: the most common unit, where a full circle measures 360 degrees.
• Radians: based on the circle's radius, offering a natural and proportional approach. Here, a full circle is \(2\pi\) radians.

To switch between these units, use the conversion:

• \(180\) degrees = \(\pi\) radians
• \(1\) degree = \(\frac{\pi}{180}\) radians

Angles can also be described as positive or negative, where positive angles rotate counter-clockwise from the positive x-axis and negative rotate clockwise.

Mastering how angles work and are measured creates a strong foundation for solving trigonometric problems, especially when dealing with inverse functions and unit circle analysis. Understanding both degrees and radians can expand your trigonometric abilities significantly.