Problem 61 - Precalculus Essentials

Question

Use a sketch to find the exact value of each expression. $$\cos \left[\tan ^{-1}\left(-\frac{2}{3}\right)\right]$$

Step-by-Step Solution

Answer

The exact value of \(\cos \left[\tan ^{-1}\left(-\frac{2}{3}\right)\right]\) is \(3\sqrt{13} / 13\).

Step 1: Sketch a right triangle

Since we're given the arctan of -2/3, this suggests a right triangle with opposite side -2 and adjacent side 3. Draw a right triangle with these sides.

Step 2: Calculate the hypotenuse

Use the Pythagorean theorem to find the hypotenuse of the triangle. The Pythagorean theorem is \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides. In this case, the hypotenuse is \(\sqrt{(3)^2 + (-2)^2} = \sqrt{13}\).

Step 3: Compute the cos value

Cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. For this triangle, this means the cosine value is \(3 / \sqrt{13}\). To rationalize this, we multiply the top and bottom by \(\sqrt{13}\) giving us \(3\sqrt{13} / 13\).

Key Concepts

Right Triangle

A right triangle is a type of triangle where one of the angles measures exactly 90 degrees, also known as a right angle. These triangles are special because they follow specific properties that help in solving problems related to angles and sides. Here’s what you need to know about them:

• The side opposite the right angle is called the hypotenuse, which is the longest side of the triangle.
• The other two sides are known as the legs and they form the right angle.
• The relationship between the sides allows us to use trigonometric functions to find missing side lengths or angles.

In the given problem, the right triangle is sketched to help visualize the relationships between the angles and sides involved. The sides given are -2 for the opposite side and 3 for the adjacent side. The hypotenuse is calculated using the Pythagorean theorem, which we’ll cover next.

Pythagorean Theorem

The Pythagorean theorem is a critical concept for working with right triangles. Formulated as \[ a^2 + b^2 = c^2 \]it states that the square of the hypotenuse (c), is equal to the sum of the squares of the other two sides (a and b). This theorem is fundamental for finding the unknown side when two sides are known.

In our problem's context, to find the hypotenuse of the triangle with sides -2 and 3, we employ the Pythagorean theorem:
- The length of the hypotenuse is calculated by substituting into the formula: \[ ( a = 3, b = -2) = c^2 = 3^2 + (-2)^2 = 9 + 4 = 13 \]- Therefore, the hypotenuse equals \( \sqrt{13} \), emphasizing the theorem’s application to find unknown dimensions in a right triangle.
This understanding helps deepen comprehension of how these geometric principles work together to solve real-world and abstract problems.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angle measures when side lengths are known. They are essential for working backwards from trigonometric ratios to angles.

Here’s how it connects to our exercise:

• Tangent and Inverse Tangent: Given \( \tan^{-1}(-\frac{2}{3}) \), you are asked to find the angle whose tangent is \(-\frac{2}{3}\). This involves drawing a triangle reflecting these ratio aspects, with the opposite being -2 and the adjacent 3.
• Calculating Cosine: With the inverse tangent providing the angle, cosine of the angle \( \theta \) is the ratio of the adjacent side to the hypotenuse, calculated as \( \cos(\theta) = \frac{3}{\sqrt{13}}\). Rationalizing gives \( \frac{3\sqrt{13}}{13} \).

Inverse trigonometric functions allow transitioning from ratio perspectives to angle considerations, expanding how right triangles and trigonometric concepts are interconnected, enhancing better problem-solving skills.