Problem 10 - Contemporary Precalculus

Question

Find the exact functional value without using a calculator: $$\sin ^{-1}(\sqrt{3} / 2)$$

Step-by-Step Solution

Answer

Answer: The exact value of the inverse sine function, sin^(-1)(sqrt(3)/2), is 60° or π/3 radians.

Step 1: Recall special angles and their sine values

The inverse sine function, sin^(-1)(x), returns the angle whose sine is x. To solve this problem, we should recall the special angles and their sine values. In particular, we need to know the sine values for angles of 30°, 45°, and 60° (or π/6, π/4, and π/3 radians, respectively). These values are: $$ \sin(30°) = \sin(\frac{\pi}{6}) = \frac{1}{2}\\ \sin(45°) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\\ \sin(60°) = \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} $$

Step 2: Identify the angle whose sine matches the given value

We are given the value \(\sin ^{-1}(\frac{\sqrt{3}}{2})\). Comparing this value with the sine values we recalled in step 1, we can see that this value matches the sine value for 60° (or π/3 radians), i.e.: $$ \sin (60°) = \sin (\frac{\pi}{3}) = \frac{\sqrt{3}}{2} $$

Step 3: Write down the exact functional value

Since the sine value for 60° (or π/3 radians) is equal to the given value, the exact functional value for \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) is 60° (or π/3 radians). So, the solution is: $$ \sin ^{-1}\left(\frac{\sqrt{3}}{2}\right) = 60° = \frac{\pi}{3}\, \text{radians} $$

Key Concepts

Special Angles

When we talk about special angles in trigonometry, we are referring to angles that have well-known trigonometric values, making them easier to work with. These are typically the angles of 30°, 45°, and 60°, which correspond to certain key radian measures: \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\).
These angles appear often in problems involving triangles, circles, and oscillations because their sine, cosine, and tangent values are simple fractions or roots. This makes calculations more straightforward without the need for a calculator.
Recognizing these angles and their associated values is crucial for solving various trigonometry problems efficiently.

Sine Function

The sine function is a fundamental concept in trigonometry that relates an angle in a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. It is commonly written as \(\sin(\theta)\).
The sine function is periodic with a period of \(2\pi\) and is symmetric about the origin, which means it is an odd function. Its range is between -1 and 1, and it goes through zero at integer multiples of \(\pi\).
Understanding the sine function helps in determining inverse sine values, wherein for a given output \(x\), the aim is to find the angle \(\theta\) such that \(\sin(\theta) = x\). In our problem, recognizing the sine value of \(\frac{\sqrt{3}}{2}\) is tied to certain angles like 60°, making it easier to spot the inverse value.

Trigonometric Values

Trigonometric values for certain angles are key in solving direct and inverse trigonometric problems. Each angle has specific sine, cosine, and tangent values:

• \(\sin(30°) = \frac{1}{2}\)
• \(\sin(45°) = \frac{\sqrt{2}}{2}\)
• \(\sin(60°) = \frac{\sqrt{3}}{2}\)

Recognizing these values allows for quick access to solving equations without returning to calculations each time. For students, memorizing or understanding these can aid in recognizing exact inverse trigonometric results swiftly during exams or real-world applications.
This is crucial in inverse trigonometric problems where you need to find which angle corresponds to a known trigonometric value, as seen in the original exercise.

Angle Measurement in Radians

Angle measurement in radians is another fundamental concept in trigonometry. Radians are a measure of angle defined such that a full circle is \(2\pi\) radians. This allows for a seamless connection between angles and the geometry of circles.
In trigonometric contexts, radians often provide a more natural measure of angles than degrees because they reflect the radius-to-circumference relationship directly. For instance, angles of \(30°\), \(45°\), and \(60°\) correspond to \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), and \(\frac{\pi}{3}\) respectively.
Understanding radians is critical for converting between degrees and radians, grasping the periodic nature of trigonometric functions, and exploring calculus-based applications of trigonometry. In our exercise, radian measure helps us express the inverse sine result accurately, as \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) being \(\frac{\pi}{3}\) emphasizes the radian's role in inverse trig functions.