Problem 23 - Thomas Calculus

Question

Find the values in Exercises \(17-28\) $$ \sin \left(\sin ^{-1}\left(-\frac{1}{2}\right)+\cos ^{-1}\left(-\frac{1}{2}\right)\right) $$

Step-by-Step Solution

Answer

The value is 1.

Step 1: Understand the Function Inverse

Begin by recalling that the inverse function \( \sin^{-1}(x) \) gives us the angle \( \theta \) such that \( \sin(\theta) = x \). Similarly, \( \cos^{-1}(x) \) provides the angle \( \phi \) such that \( \cos(\phi) = x \). We need to determine these angles for the given values.

Step 2: Evaluate \( \sin^{-1} \left(-\frac{1}{2}\right) \)

The angle \( \theta \) for which \( \sin(\theta) = -\frac{1}{2} \) is in the fourth quadrant for \(\sin^{-1} \). Hence, \( \theta = -\frac{\pi}{6} \).

Step 3: Evaluate \( \cos^{-1} \left(-\frac{1}{2}\right) \)

The angle \( \phi \) for which \( \cos(\phi) = -\frac{1}{2} \) is \( \frac{2\pi}{3} \) which is in the second quadrant.

Step 4: Sum the Angles

Add the angles obtained: \( \theta + \phi = -\frac{\pi}{6} + \frac{2\pi}{3} \). This results in \( \frac{\pi}{2} \).

Step 5: Evaluate the Sine of the Sum

Compute \( \sin \left(\frac{\pi}{2}\right) \), which is equal to 1.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the tools that help you retrieve angles from given trigonometric ratios. When you have a ratio, the inverse function brings it back to an angle.
For example,

• The function \( \sin^{-1}(x) \) tells us the angle \( \theta \) such that \( \sin(\theta) = x \).
• Similarly, \( \cos^{-1}(x) \) provides the angle \( \phi \) such that \( \cos(\phi) = x \).

Inverse sine, or \( \sin^{-1} \), typically returns an angle in the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This means the angle is found in the first or fourth quadrant. So, for \( \sin^{-1}\left(-\frac{1}{2}\right) \), the angle is \(-\frac{\pi}{6}\), in the fourth quadrant.
On the other hand, inverse cosine, \( \cos^{-1} \), yields an angle between \(0\) and \(\pi\), which places it in the first or second quadrant. Hence, \( \cos^{-1}\left(-\frac{1}{2}\right) \) is \(\frac{2\pi}{3}\), located in the second quadrant.
These functions neatly bridge the gap between ratio and angle, allowing us to solve complex trigonometric equations.

Angle Sum Identities

Angle sum identities are valuable tools in trigonometry, essential for simplifying expressions where multiple angles are involved. They allow us to express trigonometric functions of sums of angles in terms of the functions of individual angles.
For sine, the angle sum identity is stated as:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

In our exercise solution, after finding the angles from inverse functions, we deduced:
\[ \theta = -\frac{\pi}{6} \] and \[ \phi = \frac{2\pi}{3} \]
Adding these angles gives a result of \( \frac{\pi}{2} \). This is a crucial step, as the identity simplifies our task by targeting the calculation of \( \sin\left(\frac{\pi}{2}\right) \).
This identity helps break down what might seem tough into manageable chunks, effectively showing the inner workings of trigonometric angles.

Sine Function

The sine function is a cornerstone of trigonometry, representing the ratio of the opposite side to the hypotenuse in a right-angled triangle. It is periodic and oscillates between -1 and 1.
The graph of \( \sin(\theta) \) shows how the function moves smoothly up and down as the angle \( \theta \) increases. This allows for predictions of values at various points, like at critical angles such as 0, \(\frac{\pi}{2}\), \(\pi\), etc.
When dealing with \( \sin\left(\frac{\pi}{2}\right) \), we reach a peak value of 1 on the sine wave. This peak corresponds to the maximum height of the wave, representing the positive amplitude.
Understanding the behavior of the sine function helps not only in strategic calculations like those in angle sum identities but also in visualizing the rise and fall inherent in oscillatory motion across various domains.