Problem 56 - Precalculus: A Concise Course

Question

Find the exact value of the expression. (Hint: Sketch a right triangle.) $$ \sec \left(\arcsin \frac{4}{5}\right) $$

Step-by-Step Solution

Answer

\(\sec(\arcsin\frac{4}{5}) = \frac{5}{3}\)

Step 1: Draw the triangle and assign side values

Since we are supposed to sketch a right triangle, suppose \(\alpha\) is the angle such that \( \sin \alpha = \frac{4}{5} \). Draw a right triangle with angle \(\alpha\), opposite side 4 units (since the sine is opposite / hypotenuse), and hypotenuse 5 units. The missing side length (adjacent to \(\alpha\)) needs to be determined.

Step 2: Compute the missing side

The Pythagorean theorem states that for any right triangle, the square of the length of the hypotenuse equals the sum of the squares of lengths of the other two sides. That is \(c^2=a^2+b^2\). Here, \(a\) is the side opposite to \(\alpha\) and \(c\) is the hypotenuse. Therefore, the adjacent side's length is \(b = \sqrt{c^2 - a^2} = \sqrt{5^2 - 4^2} = \sqrt{9} = 3 \) units.

Step 3: Compute the sec of the angle

The secant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the adjacent side. So, \( \sec \alpha = \frac{c}{b} = \frac{5}{3} \).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This specific angle is what makes many calculations in trigonometry possible. In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are referred to as the adjacent and opposite sides, depending on which angle you are considering apart from the right angle.

When sketching a right triangle to solve trigonometric problems, like the one in our exercise, it's crucial to identify and label these sides accurately. This helps in establishing relationships between the angles and sides using trigonometric functions.

Pythagorean Theorem

The Pythagorean theorem is a fundamental relation in geometry that applies specifically to right triangles. It is expressed as \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the lengths of the other two sides.

This theorem is valuable because it allows us to calculate the unknown length of any side, provided we know the lengths of the other two. In our exercise, knowing that the sine of an angle \( \alpha \) was \( \frac{4}{5} \), we established our right triangle with the opposite side as 4 units and the hypotenuse as 5 units.

Using the Pythagorean theorem:

• Hypotenuse (\( c \)) = 5
• Opposite (\( a \)) = 4
• Adjacent ( \( b \)) = \( \sqrt{c^2 - a^2} = \sqrt{5^2 - 4^2} \)

Thus, the adjacent side was calculated to be 3 units, completing the triangle dimensions.

Secant Function

The secant function, denoted as \( \sec \theta \), is a trigonometric function that relates the hypotenuse of a right triangle to its adjacent side for a given angle \( \theta \).

• The mathematical expression for the secant function is \( \sec \theta = \frac{1}{\cos \theta} \).
• You can also describe it as \( \sec \theta = \frac{c}{b} \), where \( c \) represents the hypotenuse, and \( b \) is the adjacent side.

In our problem, the task was to find \( \sec(\arcsin(\frac{4}{5})) \). After we solved for the sides using the right triangle, we calculated \( \sec \alpha = \frac{5}{3} \).

Understanding the secant function is key because it is part of the reciprocal trigonometric functions, which are often crucial in solving trigonometric equations and expressions, especially where angles are inverted or arc functions are used.