Problem 57 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The exact value of the given expression is \(-\sqrt{8}\).

Step 1: Find the angle using the inverse cosine function

Firstly, the inverse cosine function needs to be evaluated. The function is denoted as \(\cos^{-1}(-1/3)\). This gives us the angle whose cosine is -1/3. By definition, the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. So, if we consider a right triangle where the adjacent side is -1 and the hypotenuse is 3, we get an angle whose cosine is -1/3. The third side of this triangle can be found using the Pythagorean theorem, which gives us \(\sqrt{3^2 - (-1)^2} = \sqrt{8} = 2\sqrt{2}\).

Step 2: Find the tangent of the angle

Then, find the tangent of the angle found in previous step. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In our case, the opposite side is \(\sqrt{8}\), and the adjacent side is -1. So, the tangent of our angle is \(\tan(\cos^{-1}(-1/3)) = -\sqrt{8}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us determine angles when we know the trigonometric value. By taking the inverse, like \( \cos^{-1} \, \, \), we find the angle whose cosine equals a given number. This is vital in right triangle calculations, where angles are not directly known but must be deduced from side lengths. Inverse cosine, or arccosine, lets us calculate angles by analyzing the ratio of the adjacent side to the hypotenuse.

• Example: \( \cos^{-1}(-\frac{1}{3}) \, \, \): This expression finds the angle with a cosine of -1/3.
• Practical application: In geometry, understanding angles in terms of cosine values helps construct accurate triangle models.

Knowing how to work with inverse trigonometric functions is crucial when the direct observation of angles isn't possible, but side ratios (like those in our triangle) can be measured.

Pythagorean Theorem

The Pythagorean theorem is an essential tool for finding unknown sides of a right triangle. It's given by the famous equation \( a^2 + b^2 = c^2 \, \, \), where \( c \, \, \) is the hypotenuse, and \( a \, \, \) and \( b \, \, \) are the other two sides. This theorem links side lengths, making it possible to solve for one side if the others are known.

• In our exercise: We know the hypotenuse (3) and one leg (-1). To find the opposite side, plug into the theorem: \( \sqrt{3^2 - (-1)^2} = \sqrt{8} = 2\sqrt{2} \, \, \).
• Usefulness: The Pythagorean theorem simplifies complex calculations, turning geometry problems into algebraic solutions.

Mastering this theorem is key in many trigonometry problems, paving the way for calculating all trigonometric functions and their inverses.

Right Triangle

In trigonometry, a right triangle is one where one angle is exactly 90 degrees. The right triangle is fundamental because each of its angles and sides has a unique relationship defined by trigonometric functions.

• The sides are represented as the opposite, adjacent, and hypotenuse relative to the angle in question.
• Trigonometric functions like sine, cosine, and tangent describe these relationships: \( \sin = \frac{\text{opposite}}{\text{hypotenuse}}, \cos = \frac{\text{adjacent}}{\text{hypotenuse}}, \tan = \frac{\text{opposite}}{\text{adjacent}} \, \, \).

In our problem, it's about understanding how the angle, opposite, and adjacent sides interact as we find \( \tan(\cos^{-1}(-1/3)) \, \, \). The angle derived from the inverse cosine influences which sides represent the tangent's opposite and adjacent sides. Recognizing these patterns ensures accurate and efficient problem-solving.