Problem 47 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The given identity \( \tan(\sin^{-1} x) = \dfrac{x}{\sqrt{1- x^2}} \) is proven to be correct through the process of expressing the terms as sides of right triangles and manipulating them using the Pythagorean theorem and definitions of trigonometric functions.

Step 1: Start With the Left Side

We begin with the left side of the identity that is, \( \tan(\sin^{-1} x) \). We can consider \( \sin^{-1} x = y \), then we have \( \sin y = x \). we obtain a right triangle with the opposite side as x and hypotenuse as 1 by using the definition of sine in a right triangle.

Step 2: Calculate Adjacent Side

In a right triangle, the side adjacent to the angle in consideration (y in this case) is calculated using the Pythagorean theorem. Let's denote the adjacent side as 'b', then using Pythagorean theorem we get \( b = \sqrt{1^2 - x^2} = \sqrt{1- x^2} \).

Step 3: Calculate Tangent

We know that the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. Therefore, \( \tan y =\dfrac{x}{\sqrt{1 - x^2}} \).

Step 4: Back-substitute

Substitute \( y = \sin^{-1} x \) back into the equation. Hence, \( \tan(\sin^{-1} x) = \dfrac{x}{\sqrt{1- x^2}} \), which is the right side of the original equation. Hence the identity is verified.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse operations of the basic trigonometric functions like sine, cosine, and tangent. For example, the inverse sine function, denoted as \( \sin^{-1}(x) \), gives you an angle whose sine is \( x \).
This is particularly useful when you need to find angles in right triangles. Instead of questioning what the sine of an angle is, you find the angle itself. This forms the basis for many applications in geometry and calculus.

• Range for \( \sin^{-1}(x) \): The values of inverse sine are typically between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).
• Useful in Angles: Helps determine angles when two sides' lengths are known.
• Common Notation Variations: Sometimes written as \( \arcsin(x) \).

Understanding these, inverse trigonometric functions are vital for solving equations and verifying identities.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, relating the lengths of sides in a right triangle. Simply put, it states that the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
Mathematically, it's expressed as:
\[ a^2 + b^2 = c^2 \] In trigonometric identities, the theorem helps find unknown side lengths, ensuring clarity when determining ratios like sine, cosine, and tangent.

• Hypotenuse Identification: Vital for correct application of the theorem.
• Applications: Supports calculation of side lengths when two sides are known.
• Simplification: Often simplifies complex trigonometric problems into manageable components.

The Pythagorean theorem not only aids in verifying identities but also supports deeper explorations into spatial reasoning and geometry.

Tangent in Right Triangles

The concept of tangent in right triangles is tied to the relationship between the triangle’s sides. The tangent of an angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side. This function is an essential part of trigonometry and crucial for understanding various angles and lengths.
The formula for tangent is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]In verifying trigonometric identities, knowing how to express tangent in terms of sine and cosine is invaluable.

• Opposite/Adjacent Sides: Clarifies which sides are used in the tangent ratio.
• Paves Way for Identities: Enables conversion between different trigonometric expressions.
• Real-World Applications: Crucial for solving practical problems involving angles.

No trigonometric relationship is complete without the understanding of tangent, making it a key concept in both theoretical and applied mathematics.