Problem 49 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \tan \left(\sin^{-1} \dfrac{x - 1}{4} \right) = \dfrac{x - 1}{\sqrt{16 - (x - 1)^2}} \)

Step-by-Step Solution

Answer

The given trigonometric identity is correct. The left hand side, after simplification, becomes \( \frac{x - 1}{\sqrt{16 - (x - 1)^2}} \) which matches the right hand side.

Step 1: Simplify the Left Hand Side

The left hand side of the equation is \( \tan \left(\sin^{-1} \dfrac{x - 1}{4} \right) \). This represents the tangent of the arc sine of the ratio \( \frac{x - 1}{4} \). We know that \( \sin \theta = \frac{opposite}{hypotenuse} \). So, we can interpret this as the sine of an angle in a right triangle where the opposite side is \(x - 1\) and the hypotenuse is 4. The adjacent side would then be \( \sqrt{16 - (x - 1)^2} \) by the Pythagorean theorem. Hence, \( \tan \left(\sin^{-1} \dfrac{x - 1}{4} \right) = \frac{x - 1}{\sqrt{16 - (x - 1)^2}} \)

Step 2: Compare with Right Hand Side

The right hand side of the equation is \( \frac{x - 1}{\sqrt{16 - (x - 1)^2}} \). Comparing it with the simplified version of the left hand side, it is evident that the two sides are equal.

Key Concepts

Tangent Function

The tangent function, commonly referred to as \( \tan \), is an essential part of trigonometry. It is a function relating an angle of a right triangle to the ratio of the opposite side to the adjacent side. Simply put, it helps us figure out the steepness of a right triangle. The formula for the tangent of an angle \( \theta \) is:\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]In our problem, the tangent function is applied to the angle that we find by taking the inverse sine of a fraction. This represents a real-world situation, turning an arc sine value into a comprehensible geometric representation.
The tangent function is particularly useful when you have either the sine or cosine and need to find the other based on a given angle. Remember that tangent can range from negative to positive infinity, so understanding its behavior is crucial in many mathematical contexts.

Inverse Trigonometric Functions

Inverse trigonometric functions are the "undo" operations for standard trigonometric functions like sine, cosine, and tangent. They are used to find the angle when we have the ratio of the sides. For example, the inverse sine, written as \( \sin^{-1} \), takes a ratio (opposite/hypotenuse) and returns an angle.

• \( \sin^{-1}(x) \): gives the angle whose sine is \( x \).
• \( \cos^{-1}(x) \): gives the angle whose cosine is \( x \).
• \( \tan^{-1}(x) \): gives the angle whose tangent is \( x \).

In the given exercise, \( \sin^{-1} \frac{x - 1}{4} \) determines the angle whose sine provides the given ratio. Then, the tangent of this angle is calculated, linking the concept of inverse trigonometric functions with the tangent function. These concepts are pivotal in fields spanning from physics to engineering because they reframe complex trigonometric issues into solvable angles.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that, for a triangle with sides \( a \), \( b \), and hypotenuse \( c \):\[ a^2 + b^2 = c^2 \]This theorem allows us to find a missing side in a right triangle if we know the lengths of the other two sides.
In our problem, we use the Pythagorean theorem to determine the length of the adjacent side when given the opposite and hypotenuse. This step is crucial to find the tangent value in our identity verification, as it relies on having both the opposite and adjacent sides.

• Given: Opposite = \( x-1 \)
• Hypotenuse = 4
• Adjacent = \( \sqrt{16 - (x - 1)^2} \)

By applying this theorem, we bridge the gap between trigonometry and basic geometry, showing how they work together to solve problems effectively. Understanding and applying the Pythagorean theorem sharpen analytical skills, crucial for tackling more advanced math topics.