Problem 27 - Calculus of a Single Variable

Question

Write the expression in algebraic form. (Hint: Sketch a right triangle, as demonstrated in Example 3.) \(\sin (\operatorname{arcsec} x)\)

Step-by-Step Solution

Answer

\(\sin (\operatorname{arcsec} x) = \frac{\sqrt{x^2-1}}{x}\)

Step 1: Identify the trigonometric relationship

The arcsecant, being the inverse of the secant function, represents the angle whose secant is x. In terms of right triangles, secant is the reciprocal of cosine and represents the ratio of the length of the hypotenuse to the length of the adjacent side. Therefore, by definition of the secant function, we can represent the hypotenuse as x units and the adjacent side as 1 unit.

Step 2: Determine the length of the opposite side

According to the Pythagorean theorem in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. Therefore, the length of the opposite side can be calculated as \(\sqrt{x^2-1}\).

Step 3: Finalize the algebraic expression

The sine of an angle in a right triangle is given by the ratio of the length of the side opposite to the angle to the length of the hypotenuse. Therefore, \(\sin (\operatorname{arcsec} x)\) = \(\frac{\sqrt{x^2-1}}{x}\).

Key Concepts

Inverse Trigonometric Functions

Understanding inverse trigonometric functions is crucial for solving problems involving angles and their measures. These functions essentially reverse the typical trigonometric functions. While a regular trigonometric function takes angles and gives values, an inverse trigonometric function does the opposite: it takes a value and gives an angle.

• Arcsecant (arcsec): It is the inverse of the secant function. If secant of an angle is a value, then arcsec of that value returns the angle. This means that \(\operatorname{arcsec}(x)\) gives you the angle whose secant is \(x\).
• Understanding secant: The secant of an angle in a right triangle is the ratio of the hypotenuse to the adjacent side. Consequently, its inverse helps us find the angle when given this ratio.

Learning these concepts helps in forming a bridge between algebra and trigonometry, allowing you to express angles in terms of given values, which is highly useful in many mathematical applications.

Right Triangle

A right triangle is a fundamental concept in trigonometry and geometry. It is a triangle where one of the angles is exactly 90 degrees. This type of triangle has some unique properties which make trigonometric functions applicable.

• Sides of a right triangle: In a right triangle, the longest side opposite the right angle is called the hypotenuse. The other two sides are referred to as the adjacent and opposite sides, relative to the angle of interest.
• Trigonometric ratios: These ratios are specifically valuable in right triangles. They relate the angles of the triangle to the lengths of its sides. Sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent) all derive their fundamental importance from such triangles.

The clear relationships between the sides of a right triangle make calculations of angles and lengths much more systematic, as seen in exercises involving inverse trigonometric functions.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone principle used to solve problems in right triangles. It states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.

The formula is expressed as:\[c^2 = a^2 + b^2\]where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the other two sides.

• Application: This theorem is instrumental in finding a missing side of a right triangle if the lengths of the other two sides are known.
• Example in context: In the example of finding \(\sin(\operatorname{arcsec}(x))\), the theorem helps compute the length of the opposite side (\(\sqrt{x^2-1}\)) from the hypotenuse (\(x\)) and adjacent side (1).

Applying the Pythagorean theorem simplifies solving algebraic expressions involving trigonometric functions and supports deeper understanding of different mathematical concepts.