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Magazine Chapter 2: Problem 28
If \(\sin ^{-1}\left(\frac{5}{x}\right)+\sin ^{-1}\left(\frac{12}{x}\right)=\frac{\pi}{2}\), then find \(x\).
Short Answer
Expert verified
The value of \( x \) is \( \frac{9}{5} \).
Step by step solution
01
Apply the identity
Apply the identity \( \sin^{-1}(a) + \sin^{-1}(b) = \sin^{-1}(a\sqrt{1-b^2}+b\sqrt{1-a^2}),\) if \( a^2 + b^2 鈮� 1 \) to our equation. Replace \( a \) with \( \frac{5}{x} \), \( b \) with \( \frac{12}{x} \). Therefore:銆�銆�\( \sin^{-1}\left(\frac{5}{x}\sqrt{1-(\frac{12}{x})^2}\right)+\frac{12}{x}\sqrt{1-(\frac{5}{x})^2} = \frac{\pi}{2}\). Simplify this equation to \( \frac{5}{x}\sqrt{1-(\frac{12}{x})^2}+\frac{12}{x}\sqrt{1-(\frac{5}{x})^2} = 1\) as sine of \( \frac{\pi}{2} \) is 1.
02
Bring denominator to one side
Move denominator \( x \) in right hand side by multiplying whole equation by \( x \). This will give us - \( 5\sqrt{1-\left(\frac{12}{x}\right)^2}+12\sqrt{1-\left(\frac{5}{x}\right)^2} = x \) .
03
Square the equation
Square the entire equation to get rid of the square roots. This gives the quadratic equation: \( \{5^2*(1-(\frac{12}{x})^2) + 2*5*12\sqrt{[1-(\frac{12}{x})^2][1-(\frac{5}{x})^2]} + 12^2*(1-(\frac{5}{x})^2)} = x^2 \).
04
Simplify the equation
Simplify the quadratic equation to \(169x^{2}-306x+144=0\).
05
Solve for x
Using the quadratic formula, find the values of \( x \). The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). This yields \( x = \frac{306 \pm \sqrt{306^2-4*169*144}}{2*169} \). After computation, the results will be \( x = \frac{9}{5}\) or \( x = \frac{16}{9}\). In the original equation, these values make the denominator equal to zero for sin inverse, which is not possible. So, the valid solution is \( x = \frac{9}{5} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Identities
Trigonometric identities are like a toolset for solving complex trigonometric equations. They help simplify expressions and make seemingly complicated problems easier to solve.
In the exercise, a particular identity involving inverse trigonometric functions was used:
To utilize this identity effectively, it's crucial to understand each component:
In the exercise, a particular identity involving inverse trigonometric functions was used:
- \( \sin^{-1}(a) + \sin^{-1}(b) = \sin^{-1}(a\sqrt{1-b^2} + b\sqrt{1-a^2}) \), provided \( a^2 + b^2 \leq 1 \).
To utilize this identity effectively, it's crucial to understand each component:
- **Inverse Trigonometric Functions:** These functions reverse standard trigonometric equations. For instance, \( \sin^{-1} \) of a number tells you the angle whose sine is that number.
- **Square Root Term:** The terms \( \sqrt{1-b^2} \) and \( \sqrt{1-a^2} \) ensure that the trigonometric identity holds true under the condition that the sum of squares of \( a \) and \( b \) does not exceed 1.
Quadratic Equations
Quadratic equations are polynomial equations of degree 2 and have the general form \( ax^2 + bx + c = 0 \). Solving them often reveals important values in mathematical problems.
In step 3 of the exercise, squaring the equation led to a quadratic equation:
Key components of this formula include:
In step 3 of the exercise, squaring the equation led to a quadratic equation:
- Original quadratic equation: \( 169x^{2} - 306x + 144 = 0 \).
Key components of this formula include:
- **Root Calculation:** The term \( b^2 - 4ac \) determines the nature of the roots. If positive, two real roots exist; if zero, one real root; and if negative, complex roots appear.
- **Determining Validity:** After deriving potential solutions, the roots must be checked against the problem's original constraints. This ensures that solutions adhere to the conditions set by inverse functions.
Problem-Solving Techniques
Problem-solving in mathematics involves a blend of strategies and logical thinking to work through complex issues effectively. In this exercise, several key techniques were employed.
- **Stepwise Simplification:** Begin by breaking down the problem into smaller, more manageable parts. This involves applying trigonometric identities to reduce complexity initially.
- **Equation Transformation:** Convert trigonometric expressions into algebraic ones, such as moving from the sine equation to a quadratic form by removing square roots through squaring.
- **Checking Solutions:** After solving, always verify that the solutions satisfy the original equation's constraints. This step ensures the accuracy and validity of the solutions.
