Problem 48 - Contemporary Precalculus

Question

Find the exact functional value without using a calculator. $$\cos \left[\tan ^{-1}(3 / 7)\right]$$

Step-by-Step Solution

Answer

Question: Find the exact value of $$\cos \left[\tan ^{-1}(3 / 7)\right]$$. Answer: $$\frac{7}{\sqrt{58}}$$

Step 1: Identify the angle

Let's denote the angle $$\tan ^{-1}(3 / 7)$$ as $$\theta$$. So, we know that the tangent of this angle is equal to the given value: $$\tan(\theta) = \frac{3}{7}$$

Step 2: Construct the right triangle

We can now create a right triangle that contains this angle $$\theta$$. Since we know that in a right triangle, tangent is the ratio of the opposite side to the adjacent side, let's represent the length of the opposite side as $$a = 3$$ and the length of the adjacent side as $$b = 7$$. Now we have a right triangle with a leg of length 3 opposite the angle $$\theta$$, a leg of length 7 adjacent to $$\theta$$, and a right angle.

Step 3: Determine the length of the hypotenuse

To calculate the cosine of the angle $$\theta$$, we need to determine the length of the hypotenuse (let's denote it as $$h$$) in our right triangle. The Pythagorean theorem states that $$a^2 + b^2 = h^2$$, where $$a$$ and $$b$$ are the legs of the triangle, and $$h$$ is the length of the hypotenuse. Substitute the given values into the formula: $$(3)^2 + (7)^2 = h^2$$ $$9 + 49 = h^2$$ $$58 = h^2$$ $$h = \sqrt{58}$$

Step 4: Calculate the cosine of the angle

Recall that the cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. So, the cosine of the angle $$\theta$$ is given by: $$\cos(\theta) = \frac{b}{h}$$ Using the previous result, we can calculate the cosine of $$\theta$$: $$\cos(\theta) = \frac{7}{\sqrt{58}}$$

Step 5: Answer the original question

The original exercise asked us to find the $$\cos \left[\tan ^{-1}(3 / 7)\right]$$, which is equal to $$\cos (\theta)$$. Our result in Step 4 shows that: $$\cos \left[\tan ^{-1}(3 / 7)\right] = \frac{7}{\sqrt{58}}$$ This is our final answer, and we have found the exact functional value without using a calculator.

Key Concepts

Inverse Trigonometric Functions

When we talk about inverse trigonometric functions, we're discussing functions that reverse what the regular trigonometric functions do. For example, if the tangent function gives you the ratio of the opposite side to the adjacent side of a right triangle, the inverse tangent (also called arctan and denoted as \( \tan^{-1} \)) gives you the angle that corresponds to a particular ratio.

• Inverse Tangent Function: In the exercise, we're dealing with \( \tan^{-1}(3/7) \). This means we're looking for an angle whose tangent is \( 3/7 \).
• Output of Inverse Functions: The result of an inverse trigonometric function is an angle, and that's handy because it gives us a way to relate our angle back to a triangle.

Understanding inverse functions is essential because they allow us to work backwards, from ratios back to angles. Once we have the angle, like in this case where \( \theta = \tan^{-1}(3/7) \), we can then use that angle to find other trigonometric functions, such as cosine.

Pythagorean Theorem

A critical tool in right triangle trigonometry is the Pythagorean Theorem. Named after the ancient Greek mathematician Pythagoras, this theorem provides a relationship between the sides of a right triangle. If a triangle has one right angle, the theorem states: \[ a^2 + b^2 = h^2 \] Here, \( a \) and \( b \) are the lengths of the two legs of the triangle, and \( h \) is the hypotenuse, or the longest side of the triangle.

• Application in Exercises: In the problem, we used this theorem to find the hypotenuse \( h \) by plugging in \( a = 3 \) and \( b = 7 \).
• Calculating the Hypotenuse: With the calculations, \( h^2 = 58 \) and therefore \( h = \sqrt{58} \).

Knowing the hypotenuse is fundamental when we later need to calculate ratios of the triangle, such as in determining the cosine. The Pythagorean Theorem is like the key that unlocks the other measurements we need from just two side lengths.

Right Triangle Trigonometry

Right triangle trigonometry is about knowing the relationships between angles and sides in right triangles. This branch of mathematics involves trigonometric functions such as sine, cosine, and tangent. They are used to express these relationships and solve problems.

• Tangent Function: This is the ratio of the opposite side to the adjacent side. It's where we started since the problem involved \( \tan^{-1}(3/7) \).
• Cosine Function: This is where we finished. The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. In the final calculation, we found \( \cos(\theta) = \frac{7}{\sqrt{58}} \).

Right triangle trigonometry is like the language we use to describe these ratios. By understanding which ratio corresponds to which trigonometric function, we can link different parts of the triangle. In this case, we moved from understanding an angle (inverse tangent) to determining another ratio (cosine), all through the framework of right triangle relationships. Remember, practicing these concepts helps to deepen your understanding and confidence in tackling similar trigonometric problems.