Problem 46 - Contemporary Precalculus

Question

Find the exact functional value without using a calculator. $$\tan \left[\cos ^{-1}(8 / 9)\right]$$

Step-by-Step Solution

Answer

Answer: The exact value of the tangent of the angle with the cosine value of 8/9 is $\frac{\sqrt{17}}{8}$.

Step 1: Construct a right triangle with the given information

Since we are given the cosine value, we can use the definition of cosine in a right triangle: $$\cos{\theta} = \frac{\text{adjacent}}{\text{hypotenuse}}$$. In this case, the cosine value is 8/9, so we can say the adjacent side is 8 and the hypotenuse is 9. Now we need to find the length of the opposite side.

Step 2: Use Pythagorean theorem to find the missing side length

We have a right triangle with adjacent side length 8 and hypotenuse length 9. To find the length of the opposite side, we will use the Pythagorean theorem: $$(\text{adjacent})^2 + (\text{opposite})^2 = (\text{hypotenuse})^2$$. Plug in the values we have: $$(8)^2 + (\text{opposite})^2 = (9)^2$$. Solve for the opposite side.

Step 3: Solve for the value of the opposite side

Given the equation \((8)^2 + (\text{opposite})^2 = (9)^2\), we have: $$64 + (\text{opposite})^2 = 81$$. Subtract 64 from both sides and get: $$(\text{opposite})^2=17$$. Now take the square root of 17: $$\text{opposite}= \sqrt{17}$$.

Step 4: Write the tangent function using the triangle

Using the definition of the tangent function in a right triangle, we have: $$\tan{\theta} = \frac{\text{opposite}}{\text{adjacent}}$$. Plug in the values of opposite and adjacent sides we found earlier: $$\tan{\theta} = \frac{\sqrt{17}}{8}$$.

Step 5: Find the tangent of the angle

Now we know that \(\theta = \cos^{-1}(8/9)\). We want to find the value of $$\tan(\cos^{-1}(8/9))$$. Using the triangle, we find the tangent of the angle directly: $$\tan(\cos^{-1}(8/9)) = \frac{\sqrt{17}}{8}$$. The exact functional value without using a calculator is therefore: $$\frac{\sqrt{17}}{8}$$.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are mathematical functions that help us determine the angles when given specific trigonometric values. These functions are essentially the reverse operations of standard trigonometric functions like sine, cosine, and tangent. For example, if you know the cosine of an angle is some value, the inverse cosine function, \(\cos^{-1}\), allows us to calculate that angle.

• Function Definitions: The inverse trigonometric functions include \( an^{-1}, \, ext{or}\, ext{arctan}\), \( an\), \(\cos^{-1}, \, ext{or}\, ext{arccos}\) for cosine, and \( an\) similarly for sine.
• Domain and Range: Inverse functions have specific domains and ranges. For example, \(\cos^{-1}\) has a range from 0 to \(\pi\), corresponding to the angle measurements.

Inverse trigonometric functions are crucial in trigonometry because they allow us to work backwards from the trigonometric ratio to find angles, making them valuable for solving problems involving right triangles.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially when dealing with right triangles. This theorem provides a relationship between the sides of a right triangle, which is crucial in calculating unknown sides when certain dimensions are known.

• The Theorem: \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the two legs of the triangle, and \(c\) is the hypotenuse, the side opposite the right angle.
• Usage: It is used to find one side of the triangle if the other two are known.

For example, if you have a right triangle with known values for two sides, you can substitute these into the Pythagorean equation to find the missing side. This was exactly the process used in the original exercise to find the opposite side when the adjacent and hypotenuse sides were known, ensuring precise calculations for other trigonometric functions.

Right Triangle

A right triangle is a specific type of triangle that contains one right angle, which is exactly 90 degrees. This type of triangle has intriguing properties, especially related to trigonometry, due to its defined angle.

• Sides in a Right Triangle: It has three sides: the hypotenuse, the opposite, and the adjacent. The hypotenuse is the longest because it is opposite the right angle.
• Trigonometric Functions: The functions sine, cosine, and tangent are based on the ratios of these sides.

In mathematical problems, like the one solved above, right triangles are often leveraged to connect geometric concepts with algebra. This involves using known functions like cosine and tangent to find missing lengths or angles, providing an advantage in trigonometric problem-solving.