Problem 27 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\cot \left(\sin ^{-1} \frac{7}{9}\right)\)

Step-by-Step Solution

Answer

\( \cot(\sin^{-1}(\frac{7}{9})) \approx 0.81 \).

Step 1: Understand the Problem

We need to find the value of \( \cot(\sin^{-1}(\frac{7}{9})) \). This means we are finding the cotangent of an angle whose sine is \( \frac{7}{9} \).

Step 2: Find the Angle

Let \( \theta = \sin^{-1}\left(\frac{7}{9}\right) \). This implies that \( \sin(\theta) = \frac{7}{9} \). We must determine \( \theta \) for which sine is \( \frac{7}{9} \) before finding \( \cot(\theta) \).

Step 3: Use the Pythagorean Identity

Recall the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \). We use this to find \( \cos(\theta) \). Therefore, \( \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{7}{9}\right)^2 = 1 - \frac{49}{81} = \frac{32}{81} \).

Step 4: Calculate \(\cos(\theta)\)

\( \cos(\theta) = \sqrt{\frac{32}{81}} = \frac{\sqrt{32}}{9} = \frac{4\sqrt{2}}{9} \) where we consider the principal value of inverse trigonometric functions to determine \( \cos(\theta) \).

Step 5: Calculate \(\cot(\theta)\)

Now find \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\frac{4\sqrt{2}}{9}}{\frac{7}{9}} = \frac{4\sqrt{2}}{7} \).

Step 6: Round to the Nearest Hundredth

Compute \( \frac{4\sqrt{2}}{7} \). First, find \( 4\sqrt{2} \approx 5.656 \), then \( \frac{5.656}{7} \approx 0.808 \.\) Round this value to the nearest hundredth.

Key Concepts

Cotangent

The cotangent function, denoted as \( \cot \theta \), is one of the six fundamental trigonometric functions. It is essential for understanding relationships in right triangles and is particularly useful in various fields like physics and engineering. To visualize, cotangent is the reciprocal of the tangent function.

• If we have a right triangle, the tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side.
• The cotangent, being the reciprocal, is the ratio of the length of the adjacent side to the opposite side, represented as \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).

In our exercise, we find \( \cot(\sin^{-1}(\frac{7}{9})) \), meaning we first determine the angle \( \theta \) for which \( \sin(\theta) = \frac{7}{9} \). Once \( \theta \) is found, we can use the relationship \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) to find its value.

Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric identity that expresses an inherent property of trigonometric functions. It states that for any angle \( \theta \), the square of the sine function plus the square of the cosine function equals one: \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This identity is crucial and frequently used, especially when dealing with inverse trigonometric functions.

• Given \( \sin(\theta) = \frac{7}{9} \), you can use the identity to find \( \cos(\theta) \). First, compute the square of the sine: \( \left(\frac{7}{9}\right)^2 = \frac{49}{81} \).
• Then, \( \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{49}{81} = \frac{32}{81} \).
• Finally, by taking the square root, we obtain \( \cos(\theta) = \frac{\sqrt{32}}{9} = \frac{4\sqrt{2}}{9} \).

This identity simplifies the process of finding trigonometric values and helps in calculating cotangent after obtaining both sine and cosine values.

Angle Measures in Radians

Understanding angle measures in radians is fundamental when working with trigonometric functions, especially in calculus and higher mathematics. Radians offer a natural way of measuring angles based on the arc length of a circle.

• A radian is the measure of an angle subtended by an arc that is equal in length to the radius of the circle.
• Unlike degrees, which are based on dividing a circle into 360 parts, a full circle encompasses \( 2\pi \) radians.
• This measurement is more intuitive when analyzing the properties of trigonometric functions, such as sine and cosine, as these functions inherently work with the geometry of circles.

Using radians allows for smoother calculations and relationships in a mathematical context. In the given exercise, it's necessary to keep the angle measure in radians when finding \( \cot(\theta) \), since trigonometric function properties are most straightforward in this unit.