Problem 57 - Precalculus Essentials

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

-sqrt(8)

Step 1: Understand \(\cos^{-1}(-\frac{1}{3})\)

The expression \(\cos^{-1}(-\frac{1}{3})\) gives us an angle whose cosine is -1/3. Now, the cosine of an angle in a right triangle is calculated as the adjacent side over the hypotenuse. So, we can consider a right triangle where the adjacent side is -1 (since cosine is negative in the second and third quadrants) and the hypotenuse is 3.

Step 2: Calculate the opposite side

By the Pythagorean theorem \((\text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2)\), we find the length of the opposite side. We already know the length of adjacent side is -1 and the hypotenuse is 3. Therefore, the opposite side will be \(\sqrt{3^2 - (-1)^2} = \sqrt{8}\).

Step 3: Calculate \(\tan\)

The tangent of an angle in a right triangle is the opposite side divided by the adjacent side. By applying these values to our triangle, we find \(\tan = \frac{\sqrt{8}}{-1} = -\sqrt{8}\)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when we know the sides of a right triangle. Unlike regular trigonometric functions, they work in reverse.

• For example, \(\cos^{-1}\) takes a ratio (like \(-\frac{1}{3}\)) and gives us the angle whose cosine is that ratio.
• It's important to remember the range where these functions operate. \(\cos^{-1}\) gives angles only in the first and second quadrants, meaning our angles will be between 0 and \(\pi\).

Using inverse functions is essential for solving trigonometry problems where we know side ratios and need to find angle measures.

Pythagorean Theorem

The Pythagorean theorem is crucial for working with right triangles. It relates the three sides: the hypotenuse, the adjacent side, and the opposite side.

• The formula is \(a^2 + b^2 = c^2\),where \(c\) is the hypotenuse, while \(a\) and \(b\) are the other two sides.
• In the problem, the hypotenuse is 3, and the adjacent side is -1.
• We use the theorem to find the opposite side: \(\sqrt{3^2 - (-1)^2} = \sqrt{8}\).

This principle is a building block in geometry and is especially useful when the lengths of other sides are known.

Tangent Function

The tangent function measures how steep a line is in a right triangle. It's the ratio of the opposite side to the adjacent side.

• In this exercise, we found the opposite side to be \(\sqrt{8}\) and the adjacent side to be -1.
• The tangent of the angle is then \(-\frac{\sqrt{8}}{1} = -\sqrt{8}\).

Knowing how to calculate tangent is crucial in trigonometry, as it helps determine slopes and angles in various mathematical and real-life scenarios.

Right Triangle Trigonometry

Right triangle trigonometry is at the heart of this problem. It focuses on relationships between the angles and sides of right triangles.

• The triangle has one angle of 90 degrees, forming the basis for the trigonometric functions.
• Using properties of the right triangle, such as cosine, sine, and tangent, we can solve for unknown parts.
• Drawing a triangle based on the given information, like having adjacent side -1 and hypotenuse 3, illustrates these relationships.

Understanding right triangle trigonometry helps us bridge the gap between geometric shapes and algebraic equations, making it foundational in mathematics education.