Problem 31 - Precalculus Mathematics for Calculus

Question

Find the exact value of the expression. $$\tan \left(\sin ^{-1} \frac{12}{13}\right)$$

Step-by-Step Solution

Answer

\( \frac{12}{5} \)

Step 1: Understand the Problem

We need to find the expression \( \tan \left( \sin^{-1} \frac{12}{13} \right) \). This involves finding the tangent of an angle whose sine is \( \frac{12}{13} \).

Step 2: Use a Right Triangle

Consider a right triangle where the opposite side to angle \( \theta \) is \( 12 \) and the hypotenuse is \( 13 \), since \( \sin \theta = \frac{12}{13} \). We need to find the adjacent side using the Pythagorean theorem: \( 13^2 = 12^2 + b^2 \).

Step 3: Apply the Pythagorean Theorem

Calculate \( b^2 = 13^2 - 12^2 = 169 - 144 = 25 \), so \( b = 5 \). Thus, the adjacent side is \( 5 \).

Step 4: Find \( \tan \theta \)

Using the triangle, \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5} \).

Step 5: Confirm the Result

Since all values have been correctly calculated using the Pythagorean theorem in a valid geometric framework, \( \tan \left( \sin^{-1} \frac{12}{13} \right) = \frac{12}{5} \) is confirmed as correct.

Key Concepts

Tangent Function

The tangent function is an important concept in trigonometry, often abbreviated as \( \tan \). It relates the angles and sides of a right triangle. In a right triangle, the tangent of an angle is determined by dividing the length of the side opposite the angle by the length of the side adjacent to the angle. Mathematical expression:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

Understanding \( \tan \) is crucial because it helps us determine the slope or steepness of an angle in a triangle. In many practical scenarios, such as in engineering and physics, calculating the tangent gives insights into angles and distances that are otherwise difficult to measure. In our problem, we are using tangent to find the angle whose sine is already given as \( \frac{12}{13} \). This illustrates the crossover between different trigonometric functions, showing how they interrelate.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that defines the relationship between the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is expressed through the formula:

• \( c^2 = a^2 + b^2 \)

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. When solving the exercise, this theorem enabled us to find the missing side of the triangle, given that one side (12) and the hypotenuse (13) were known. By rearranging the theorem, \( b^2 = 13^2 - 12^2 = 169 - 144 = 25 \), we calculated using square root to find the adjacent side \( b = 5 \). This process allows us to fully solve the triangle and apply other trigonometric functions accurately.

Right Triangle

A right triangle is a triangle where one of its angles is exactly 90 degrees. This unique property allows for the application of trigonometric functions and the Pythagorean Theorem, making it a central shape in geometry and trigonometry. Right triangles consist of:

• One hypotenuse (the longest side), opposite the right angle.
• Two legs, which form the right angle.

In our exercise, the particular right triangle is constructed with sides 12, 5, and 13. The identification of this triangle, characterized by \( \sin \theta = \frac{12}{13} \), allowed geometric applications like the use of \( \tan \), cementing the relationship between the side lengths. Recognizing and accurately drawing or visualizing a right triangle serves as an essential step for solving problems involving trigonometric identities and expressions.