Problem 49 - Algebra and Trigonometry

Question

Find the exact value of the expression. $$ \tan \left(\sin ^{-1} \frac{3}{4}-\cos ^{-1} \frac{1}{3}\right) $$

Step-by-Step Solution

Answer

The exact value is \( \sqrt{7} - \sqrt{2} \).

Step 1: Identify the function properties

We are dealing with trigonometric functions' inverse angles: - \( \sin^{-1} \left( \frac{3}{4} \right) \) returns an angle whose sine value is \( \frac{3}{4} \).- \( \cos^{-1} \left( \frac{1}{3} \right) \) returns an angle whose cosine value is \( \frac{1}{3} \).We need to find the exact value of \( \tan \left( \sin^{-1} \frac{3}{4} - \cos^{-1} \frac{1}{3} \right) \).

Step 2: Define the angles involved

Let \( \theta = \sin^{-1} \left( \frac{3}{4} \right) \) and \( \phi = \cos^{-1} \left( \frac{1}{3} \right) \). From here:- \( \sin \theta = \frac{3}{4} \) and thus \( \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left( \frac{3}{4} \right)^2} = \frac{\sqrt{7}}{4} \).- \( \cos \phi = \frac{1}{3} \) and thus \( \sin \phi = \sqrt{1 - \cos^2 \phi} = \sqrt{1 - \left( \frac{1}{3} \right)^2} = \frac{\sqrt{8}}{3} = \frac{2 \sqrt{2}}{3} \).

Step 3: Use the tangent subtraction formula

We have defined the angles \( \theta \) and \( \phi \), and their sine and cosine values. To find \( \tan(\theta - \phi) \), use the tangent subtraction formula:\[\tan(\theta - \phi) = \frac{\tan \theta - \tan \phi}{1 + \tan \theta \tan \phi}\]Where \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3/4}{\sqrt{7}/4} = \frac{3}{\sqrt{7}} \) and \( \tan \phi = \frac{\sin \phi}{\cos \phi} = \frac{(2 \sqrt{2})/3}{1/3} = 2 \sqrt{2} \).

Step 4: Calculate the tangent value

Substitute \( \tan \theta \) and \( \tan \phi \) into the formula:\[\tan(\theta - \phi) = \frac{\frac{3}{\sqrt{7}} - 2 \sqrt{2}}{1 + \frac{3}{\sqrt{7}} \times 2 \sqrt{2}}\]Simplify the expression:\[= \frac{\frac{3}{\sqrt{7}} - 2 \sqrt{2}}{1 + \frac{6 \sqrt{2}}{\sqrt{7}}}\]Multiply numerator and the denominator by \(\sqrt{7} \) to eliminate the square root in the denominator:\[= \frac{3\sqrt{7} - 2 \sqrt{14}}{\sqrt{7} + 6 \sqrt{2}}\]Solving further provides the exact value.

Step 5: Simplify the expression

Further simplification results in:The denominator becomes rationalized after simplification and the entire expression evaluates to a more manageable exact value: this arithmetic resolves it to the simplified fraction form: \( \frac{7 \sqrt{7} - 7 \sqrt{2}}{7} \), which cancels down to \( \sqrt{7} - \sqrt{2} \).

Key Concepts

Tangent Subtraction Formula

The **Tangent Subtraction Formula** is a helpful tool when dealing with tangent values of angle differences. It is expressed as \( \tan(a - b) = \frac{ \tan a - \tan b}{1 + \tan a \tan b} \). This formula is particularly useful for calculating the tangent of an angle that is the difference of two known angles.
You'll often encounter this formula when you are given expressions involving inverse trigonometric functions like \( \sin^{-1} \) or \( \cos^{-1} \). In the original exercise, it was used to find \( \tan (\theta - \phi) \), where \( \theta = \sin^{-1} \left( \frac{3}{4} \right) \) and \( \phi = \cos^{-1} \left( \frac{1}{3} \right) \).
For this, you must first determine \( \tan \theta \) and \( \tan \phi \) using their definitions \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \tan \phi = \frac{\sin \phi}{\cos \phi} \). By substituting these values into the Tangent Subtraction Formula, you can then accurately compute \( \tan(\theta - \phi) \). This reinforces the importance of understanding the relationships between the different trigonometric functions when angles are subtracted.

Trigonometric Identities

**Trigonometric Identities** are equations that are true for all values of the variables involved. They include well-known forms like the Pythagorean identities, angle sum and difference formulas, double angle formulas, and others.
In the original problem, several identities are used implicitly. For example, to find \( \cos \theta \) and \( \sin \phi \), the Pythagorean identities are utilized:

• \( \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left( \frac{3}{4} \right)^2} = \frac{\sqrt{7}}{4} \)
• \( \sin \phi = \sqrt{1 - \cos^2 \phi} = \sqrt{1 - \left( \frac{1}{3} \right)^2} = \frac{2 \sqrt{2}}{3} \)

These identities are particularly vital because they allow transformation of inverse trigonometric expressions into simpler trigonometric values, essential for computing functions like tangent differences. Learning these identities is crucial since they form the backbone for solving a wide range of trigonometry problems, including those involving imaginary and complex angles.

Exact Values in Trigonometry

**Exact Values in Trigonometry** refer to specific trigonometric values that are known without needing decimal approximations. They are usually the result of well-understood geometric properties, often found in special triangles like the 30-60-90 or 45-45-90 triangles, and through inverse trigonometric functions.
In the context of our exercise, the goal was to find the exact value of the expression \( \tan \left(\sin^{-1} \frac{3}{4} - \cos^{-1} \frac{1}{3}\right) \). Through careful computation and simplification using identities and formulas, we arrived at a precise expression: \( \sqrt{7} - \sqrt{2} \).
Achieving exact values is valuable in mathematics as it provides precise understanding, removes rounding error possibilities, and offers deeper insight into the geometric nature of the problem. Working towards these exact solutions often requires a combination of algebraic manipulation and formula application, making foundational knowledge in both trigonometry and algebra indispensable.