Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\sec \left(\arcsin \frac{4}{5}\right)\)

Step-by-Step Solution

Answer

The exact value of \( \sec \left(\arcsin \frac{4}{5}\right) \) is \( \frac{5}{3} \).

Step 1: Draw and measure

To find the exact value of \( \sec(\arcsin \frac{4}{5}) \), sketch a right triangle with an angle \( \theta \) such that \( \sin(\theta)=\frac{4}{5} \). If \( \sin(\theta)=\frac{4}{5} \), then \( \theta = \arcsin(\frac{4}{5}) \). In other words, \( \theta \) is an angle whose sine equals \( \frac{4}{5} \). You can represent the \( \frac{4}{5} \) as the ratio of the opposite side to the hypotenuse of the right triangle, so make opposite side as 4 (opposite to angle \( \theta \)) and hypotenuse as 5. You do not know the adjacent side yet, call it 'a' for now.

Step 2: Find the adjacent side

Use the Pythagorean theorem to find the length of the adjacent side 'a'. The Pythagorean theorem states that for a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, this gives us: \( a = \sqrt{5^2 - 4^2 } = 3 \)

Step 3: Find \( \sec(\theta) \) value

Secant of an angle \( \theta \) in a right triangle is the ratio of the length of the hypotenuse to the length of the adjacent side. So \( \sec(\theta) = \frac{Hypotenuse}{Adjacent} = \frac{5}{3} \). So, \( \sec (\arcsin \frac{4}{5}) = \frac{5}{3} \)

Step 4: Verify with Graphing Utility

Use a graph representation of the secant and arcsin function to check your answer. The result will match confirming our hand-made calculations.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It's closely related to the cosine function and is defined as the reciprocal of the cosine of an angle in a right triangle. This means that for any angle \( \theta \), the secant function can be expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

In a right triangle, the secant of an angle is the ratio of the length of the hypotenuse to the length of the adjacent side. Understanding this relationship is crucial when working with trigonometric functions and solving related problems. The secant function is especially useful when dealing with angles where the cosine may be zero, given its reciprocal nature. That is why the secant function is undefined for angles where the cosine value is zero.
The behavior of the secant function can be observed graphically as well, where it shares periodic properties with the cosine. Its graph has discontinuities (asymptotes) wherever the cosine graph crosses zero.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics for finding angles when given trigonometric ratios. These functions essentially "undo" the primary trigonometric functions. For example, the inverse sine, or \( \arcsin \), function takes a ratio and returns the corresponding angle:

• \( \arcsin(x) = \theta \)

Here, \( \theta \) is the angle whose sine is \( x \). Similarly, there are other inverse trigonometric functions such as \( \arccos \) and \( \arctan \). These functions are crucial when solving equations that involve trigonometric expressions, by allowing us to find specific angle measures.
Keep in mind that inverse trigonometric functions have restricted domains and ranges so they can function uniquely. For example, \( \arcsin \) is typically defined for \( x \) values between \(-1\) and \(1\) and returns angle values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). This ensures that every input corresponds to just one output, preserving their function status.

Right Triangle

A right triangle is a crucial concept in trigonometry, characterized by having one angle that measures exactly 90 degrees, known as the right angle. The longest side opposite this right angle is called the hypotenuse. The other two sides are called the opposite and adjacent sides relative to the angle of interest.To solve the exercise, we first sketched such a triangle based on the information provided from the trigonometric identity \( \sin(\theta) = \frac{4}{5} \). This implies that in the right triangle, the side opposite the angle \( \theta \) is 4 units long, and the hypotenuse is 5 units long.
We calculated the length of the adjacent side using the Pythagorean theorem, an essential tool when working with right triangles:

• The formula is \( a^2 + b^2 = c^2 \)

Here, \( a \) corresponds to the adjacent side, \( b \) to the opposite side, and \( c \) to the hypotenuse. In our exercise, this gave us an adjacent side length of 3, confirming our sketch and calculations.Understanding right triangles allows us to explore and solve trigonometric problems efficiently and gain deep insight into geometric relationships.