Problem 69 - 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.) \(\sin \left[\arccos \left(-\frac{2}{3}\right)\right]\)

Step-by-Step Solution

Answer

The exact value of the expression \(\sin \left[\arccos \left(-\frac{2}{3}\right)\right]\) is \(\frac{\sqrt{5}}{3}\).

Step 1: Understand inverse cosine

Inverse cosine, denoted \(\arccos\), is the function that returns the angle whose cosine is the given number. In this case, we are given \(\arccos(-\frac{2}{3})\). This represents an angle \(θ\), where \(-\frac{2}{3}\) is the cosine of \(θ\). Since the cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse, we can express \(θ\) as having an adjacent side of -2 (since cosine is negative in the second and third quadrants) and a hypotenuse of 3.

Step 2: Find missing side using Pythagorean theorem

In a right triangle, the Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. We already know the lengths of the adjacent side and the hypotenuse. Let's denote the missing side (opposite side to angle \(θ\)) as \(x\). Then by the Pythagorean theorem: \[x = \sqrt{3^2 - (-2)^2} = \sqrt{9 - 4} = \sqrt{5}\]

Step 3: Calculate sine of the angle

Now that we have all sides of the right triangle, we can find the sine of the angle \(θ\). The sine of an angle in a right triangle is defined as the ratio of the opposite side to the hypotenuse. So: \[\sin(θ) = \frac{x}{\text{Hypotenuse}} = \frac{\sqrt{5}}{3}\]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies to right triangles. It's a powerful tool for figuring out the relationship between the sides of a right triangle. This theorem states:
\[ a^2 + b^2 = c^2 \] where \(a\) and \(b\) are the lengths of the legs (the two shorter sides) of the triangle, and \(c\) is the length of the hypotenuse (the longest side, opposite the right angle).
The theorem is incredibly useful since it only requires two known sides to find the third.
In the exercise, we know the hypotenuse and one leg, which allows us to rearrange the theorem to find the missing leg. The **hypotenuse** is 3, and the **adjacent side** is -2. Thus, by replacing \(a, b,\) and \(c\) with real numbers, you get the equation:
\[ x = \sqrt{c^2 - a^2} = \sqrt{3^2 - (-2)^2} = \sqrt{9 - 4} = \sqrt{5} \]
This result gives the length of the opposite side as \(\sqrt{5}\). It helps to visualize each step, making the Pythagorean Theorem much easier to understand.

Right Triangle

A right triangle, as the name suggests, possesses one right angle (exactly 90 degrees). The other two angles are always less than 90 degrees. This specific structure is critical for creating trigonometric ratios and applying the Pythagorean Theorem.
The sides of a right triangle include:

• The **hypotenuse**: longest side, opposite the right angle.
• The **adjacent side**: close to the angle of interest other than the right angle.
• The **opposite side**: opposite the angle other than the right angle.

This configuration allows each side to relate through trigonometric ratios. In our previous discussion, the exercise attempts to solve a triangle where these relationships are explored.
The idea is to first use known ratios, such as cosine, to reconstruct the side properties of the triangle. Once you have these, you can comfortably navigate other trigonometric functions like sine and tangent.

Trigonometric Ratios

Trigonometric ratios come from the study of right triangles, helping us connect angles and sides. For an angle \(\theta\) in a right triangle:

• **Sine** \((\sin)\): ratio of the opposite side to the hypotenuse. \[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
• **Cosine** \((\cos)\): ratio of the adjacent side to the hypotenuse. \[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
• **Tangent** \((\tan)\): ratio of the opposite side to the adjacent side. \[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]

In the example problem, you needed to find the sine of an angle given its cosine value. Since \(\cos(\theta) = -\frac{2}{3}\), the task was to find \(\sin(\theta)\). By using the Pythagorean identity, which relates sine and cosine, you reverse-engineer it's possible to find the missing side. Once the correct side lengths are known, the sine ratio allows calculation:
\[ \sin(\theta) = \frac{\sqrt{5}}{3} \]
Understanding these ratios makes it easier to deal with trigonometric inverses and their applications, providing a firm grasp of trigonometry.