Problem 114 - Algebra and Trigonometry

Question

Use a sketch to find the exact value of \(\cos \left(\tan ^{-1} \frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The exact value of \( \cos(\tan^{-1}(\frac{3}{4})) \) is \( \frac{4}{5} \)

Step 1: Identify the Right Triangle

First let's identify a right triangle with this property: The opposite side is 3 and the adjacent side is 4. This can be done by dividing a right triangle's legs by the greatest common factor.

Step 2: Calculate the Hypotenuse

To find out the hypotenuse, apply Pythagorean theorem. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So \( c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = 5 \)

Step 3: Compute the Cosine

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. So for the angle \( \theta = \tan^{-1}(\frac{3}{4}) \), it will look like this: \( \cos(\theta) = \frac{adjacent}{hypotenuse} = \frac{4}{5} \)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are incredibly useful when we need to find an angle given a ratio of sides in a right triangle. They "undo" regular trigonometric functions, helping us determine angles. The inverse tangent function, written as \( \tan^{-1} \) or "arctan," specifically helps us find an angle when you know the ratio of the opposite side to the adjacent side in a right triangle.
In our problem, we have \( \tan^{-1}\left(\frac{3}{4}\right)\). This means we need to find an angle \( \theta \) such that the tangent of \( \theta \) is \( \frac{3}{4} \). By sketching a right triangle, where the opposite side is 3 units and adjacent side is 4 units, we can visualize this ratio. The angle \( \theta \) computed here is precisely the angle whose tangent would be \( \frac{3}{4} \).
Inverse trigonometric functions are often used in fields like engineering, physics, and even computer graphics, where resolving component angles from vectors is necessary.

Right Triangles

A right triangle is a special type of triangle where one of the angles is precisely \( 90^\circ \), or a right angle. This makes it the primary focus of trigonometric studies. A right triangle consists of three sides: the hypotenuse, which is the longest side, and the other two sides known as legs.
In our example, with the tan inverse function involved, we begin with identifying side lengths based on the ratio provided. This gives us a right triangle having legs of 3 and 4. The angle \( \theta = \tan^{-1}(\frac{3}{4}) \) is formed between the hypotenuse and the adjacent side.
Right triangles are pivotal in many applications. For instance, they are used in navigation, architecture, and determining distances that aren't directly measurable.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of sides in a right triangle. It's famously stated as \( a^2 + b^2 = c^2 \), where \( c \) is the length of the hypotenuse and \( a \) and \( b \) are the lengths of the other two sides.
In the provided problem, we use the Pythagorean theorem to calculate the hypotenuse of our triangle whose legs are 3 and 4. So, substituting the values, we have \( c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \). This gives us the precise value of the hypotenuse needed for further calculations, like determining cosine.
The Pythagorean theorem is more than just a formula; it’s a gateway to comprehending distances and proving properties in various geometrical settings and constructing reliable designs in engineering contexts.