Problem 32 - Algebra and Trigonometry

Question

Find the exact value of the expression. $$ \cot \left(\sin ^{-1} \frac{2}{3}\right) $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{5}}{2} \)

Step 1: Understand the Problem

We need to find the value of \( \cot \left(\sin^{-1} \frac{2}{3}\right) \). This involves finding the cotangent of an angle whose sine is \( \frac{2}{3} \).

Step 2: Use Identity for Sine

Since \( \sin(\theta) = \frac{2}{3} \), we can represent this using the sides of a right triangle: the opposite side is 2, and the hypotenuse is 3. We need to find the adjacent side to find \( \cot(\theta) \).

Step 3: Pythagorean Theorem

To find the adjacent side, use the Pythagorean Theorem: \[ (\text{adjacent})^2 + (\text{opposite})^2 = (\text{hypotenuse})^2 \] \[ a^2 + 2^2 = 3^2 \] \[ a^2 + 4 = 9 \] \[ a^2 = 5 \] \[ a = \sqrt{5} \] Thus, the adjacent side is \( \sqrt{5} \).

Step 4: Calculate Cotangent

Now, \( \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} \). Substitute the values:\[ \cot(\theta) = \frac{\sqrt{5}}{2} \]

Step 5: Conclusion

We found that \( \cot \left(\sin^{-1} \frac{2}{3}\right) = \frac{\sqrt{5}}{2} \).

Key Concepts

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions are vital tools in mathematics, allowing us to find angles when we know the ratio of sides in a right triangle. These functions essentially reverse the process of regular trigonometric functions like sine, cosine, and tangent. For example, the function \( \sin^{-1} \) is the inverse of the sine function. It tells us the angle whose sine is a given number.

• These functions are only defined for certain ranges of angles. For instance, \( \sin^{-1} \) typically results in an angle between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• When you see \( \sin^{-1} \frac{2}{3} \), it means you're looking for an angle \( \theta \) such that \( \sin(\theta) = \frac{2}{3} \).

This angle is used in further calculations, such as determining other trigonometric values of that angle.

Applying the Pythagorean Theorem

The Pythagorean Theorem is a cornerstone in geometry, helping us relate the sides of a right triangle. It is expressed as \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs, and \( c \) is the hypotenuse.

• In our problem, since the sine of angle \( \theta \) is \( \frac{2}{3} \), we know the opposite side is 2 and the hypotenuse is 3.
• To find the adjacent side, we rearrange the theorem to \( a^2 = c^2 - b^2 \), substitute \( 3^2 - 2^2 \), solve to get \( a = \sqrt{5} \).

Finding this adjacent side is crucial as it lets us compute the cotangent of the angle. It showcases the beauty of the Pythagorean Theorem in connecting geometric concepts.

The Role of Cotangent in Trigonometry

Cotangent, often abbreviated as \( \cot \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function and relates to a right triangle's sides as \( \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} \).

• In our specific case, since we know \( \sin^{-1} \frac{2}{3} \) gives us an angle \( \theta \) where the opposite side is 2 and the adjacent side is \( \sqrt{5} \), we compute \( \cot(\theta) \).
• Substituting the values, \( \cot(\theta) = \frac{\sqrt{5}}{2} \), helping us find the exact value needed.

This demonstrates how cotangent, combined with inverse functions and the Pythagorean theorem, aids in solving trigonometric equations by using known triangle dimensions.