Problem 27 - Algebra and Trigonometry

Question

Find the exact value of the expression. $$ \sin \left(\cos ^{-1} \frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The exact value is \( \frac{4}{5} \).

Step 1: Understand the Inverse Cosine Function

The expression involves \( \cos^{-1} \left( \frac{3}{5} \right) \), which means we are looking for an angle \( \theta \) such that \( \cos(\theta) = \frac{3}{5} \).

Step 2: Use the Pythagorean Identity

From the Pythagorean Identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), we substitute \( \cos(\theta) = \frac{3}{5} \) to find \( \sin(\theta) \).

Step 3: Calculate \( \sin(\theta) \)

Solve for \( \sin(\theta) \) using the identity:\[\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}\]Thus, \( \sin(\theta) = \sqrt{\frac{16}{25}} = \frac{4}{5} \) or \( \sin(\theta) = -\frac{4}{5} \).

Step 4: Determine the Correct Sign

Since \( \cos^{-1} \) outputs angles in the range \([0, \pi]\), where the sine function is non-negative, we have \( \sin(\theta) = \frac{4}{5} \).

Step 5: Conclusion

We find that \( \sin(\cos^{-1} (\frac{3}{5})) = \frac{4}{5} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are important tools for finding angles when the value of a trigonometric ratio is known. When we talk about \( \cos^{-1}(x) \), it cues us to search for an angle \( \theta \) such that the cosine of \( \theta \) equals \( x \). This is particularly helpful when dealing with problems requiring the recalculation of dimensions in right-angled triangles.

• Understanding \( \cos^{-1}(x) \): Here, \( \cos^{-1}(\frac{3}{5}) \) means we look for an angle \( \theta \) satisfying \( \cos(\theta) = \frac{3}{5} \).
• Angle Range: \( \cos^{-1}(x) \) yields angles within the range \([0, \pi]\), which means we are dealing with angles in the first two quadrants of the unit circle.
• Determining Other Trigonometric Ratios: Knowing \( \theta \) from an inverse trigonometric function allows us to find other ratios like \( \sin(\theta) \) using identities.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry that elegantly links sine and cosine functions. It states: \[\sin^2(\theta) + \cos^2(\theta) = 1\]This relationship holds for any angle \( \theta \). So, if you know one of the trigonometric functions, you can easily find the other.

• In our example, substituting \( \cos(\theta) = \frac{3}{5} \), we have: \( \sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \left(\frac{3}{5}\right)^2 = \frac{16}{25} \)
• Since we are solving \( \sin^2(\theta) = \frac{16}{25} \), by taking the square root, we find: \( \sin(\theta) = \pm \frac{4}{5} \)

When taking a square root, remember the value can be positive or negative. However, the context and range of the function (\( [0, \pi] \) for cosine) usually help determine the correct sign.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for each angle. They are essential for simplifying expressions and solving equations. Here are a few key types of identities that show the relationships between the main trigonometric functions:

• Reciprocal Identities: These wonder identities allow you to express the basic functions in terms of their reciprocals. For example, \( \sin(\theta) = \frac{1}{\csc(\theta)} \).
• Ratio Identities: These tell us the ratios between different functions, such as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
• Pythagorean Identities: Apart from the primary \( \sin^2(\theta) + \cos^2(\theta) = 1 \), there are variations like \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

These identities help in solving complex trigonometric equations, finding angle values, and transforming expressions to a more digestible form. When approached correctly, they make understanding trigonometric exercises like finding \( \sin(\cos^{-1}(\frac{3}{5})) \) much clearer and more approachable.