Problem 72 - Algebra and Trigonometry

Question

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$ \cot \left(\sin ^{-1} \frac{\sqrt{x^{2}-9}}{x}\right) $$

Step-by-Step Solution

Answer

So, \(\cot(\sin^{-1} \frac{\sqrt{x^{2}-9}}{x})\) can be expressed as the algebraic expression \(\frac{3}{\sqrt{x^{2}-9}}\)

Step 1: Understanding the Inverse Sine Function

For the expression \(\frac{\sqrt{x^{2}-9}}{x}\), consider \(\sin^{-1}(\frac{\sqrt{x^{2}-9}}{x})\) as an angle \(\theta\), in a right-angled triangle. By the definition of the sine function, \(\sin{\theta} = \frac{opposite}{hypotenuse}\). Thus, \(\frac{\sqrt{x^{2}-9}}{x}\) represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.

Step 2: Draw the Triangle

Draw a right-angled triangle with the opposite side length as \(\sqrt{x^{2}-9}\) and the hypotenuse as \(x\). The adjacent side can be determined using the Pythagorean theorem as \(x^{2}- (\sqrt{x^{2}-9})^{2}\), which equals to 9. So, the adjacent side has length 3.

Step 3: Evaluate the Original Expression

The original expression is \(\cot(\sin^{-1} \frac{\sqrt{x^{2}-9}}{x})\), i.e., the cotangent of angle \(\theta\). In the right-angled triangle, the cotangent of an angle is defined as the ratio of the length of the adjacent side to the opposite side. Given that the adjacent side is 3 and the opposite side is \(\sqrt{x^{2}-9}\), the original expression simplifies to \(\frac{3}{\sqrt{x^{2}-9}}\).

Key Concepts

Cotangent of an Angle

The cotangent of an angle in a right triangle is a fundamental trigonometric function that relates the lengths of sides in a right triangle. Specifically, cotangent is defined as the ratio of the length of the adjacent side (next to the angle) to the length of the opposite side (across from the angle). To symbolize this, we use the following equation: \[\begin{equation}\cot(\theta) = \frac{{\text{{adjacent}}}}{{\text{{opposite}}}}\end{equation}\]In trigonometric functions like sine and cotangent, it's crucial to understand that they are based on the angles and dimensions of a right triangle. When dealing with an exercise involving the cotangent function, think visually: imagine or even draw the triangle to help you conceptualize the problem. Always list down what you know: the lengths of sides or the measure of angles. From the text, we're given that the adjacent side length is 3 and the opposite side is \(\sqrt{x^{2}-9}\). Thus, the cotangent of \(\theta\) is simply the ratio \(\frac{3}{\sqrt{x^{2}-9}}\).As an education tip, remember that the inverse trigonometric functions 'undo' what the regular trigonometric functions do. They allocate an angle from a known ratio of sides. So, by stating \(\cot(\sin^{-1}(\frac{\sqrt{x^{2}-9}}{x}))\), we're really seeking the angle whose sine gives us that ratio, and then finding the cotangent of that same angle.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, especially when solving problems involving right triangles. Simply put, this theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is expressed as:\[\begin{equation}a^{2} + b^{2} = c^{2}\end{equation}\]where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the other two sides. Utilizing the Pythagorean theorem in our problem helps us find the length of the adjacent side after knowing the hypotenuse and opposite side lengths. In this exercise, once we've determined \(sin^{-1}(\frac{\sqrt{x^{2}-9}}{x})\) as \(\theta\), the length of the hypotenuse (x) and opposite side (\(\sqrt{x^{2}-9}\)) are given. To find the adjacent side, we set up the equation \(x^2 = 3^2 + (\sqrt{x^{2}-9})^2\), ultimately simplifying to \(9\).Remember to encourage students to familiarize themselves with squares and square roots, as they frequently appear in using the Pythagorean theorem. Having a clear step-by-step approach to solving for unknowns using this theorem is also crucial for mastering problems involving right triangles.

Sine Function

Understanding Sine in a Right Triangle

The sine function is another foundational concept in trigonometry and is defined within the context of a right triangle. For an angle \(\theta\), sine (written as \(\sin\)) is the ratio of the length of the side opposite to \(\theta\) to the length of the hypotenuse. The formula to express this is:\[\begin{equation}\sin(\theta) = \frac{{\text{{opposite}}}}{{\text{{hypotenuse}}}}\end{equation}\]When provided with an inverse sine function, like \(\sin^{-1}\), our goal is to determine the angle whose sine is the given ratio. Therefore, finding \(\sin^{-1}(\frac{\sqrt{x^{2}-9}}{x})\) implies that we're looking for an angle \(\theta\) where \(\sin(\theta)\) equals \(\frac{\sqrt{x^{2}-9}}{x}\). Once \(\theta\) is identified, we can then refer back to our right triangle to find other trigonometric ratios, such as cotangent, by using the lengths of the triangle's sides.When explaining the sine function to students, emphasize visualization and the use of known information to solve for the unknown. Encouraging the practice of drawing right triangles can significantly help learners to picture which sides are involved in the ratio that defines the sine of an angle. Equally, explaining the concept of inverse functions will aid their comprehension of how an angle can be 'retrieved' from the function's value.