Problem 7 - Precalculus

Question

Find \(\sin \theta\) and \(\cos \theta,\) where \(\theta\) is the (acute) angle of rotation that eliminates the \(x^{\prime} y^{\prime}\) -term. Note: You are not asked to graph the equation. $$25 x^{2}-24 x y+18 y^{2}+1=0$$

Step-by-Step Solution

Answer

\(\theta\) is calculated from \(\tan 2\theta = \frac{-24}{7}\). Use \(\theta\) to find \(\sin \theta\) and \(\cos \theta\).

Step 1: Identify the Coefficients

The given quadratic equation is of the form \(ax^2 + bxy + cy^2 + d = 0\). By comparing it with the generic form: \(ax^2 + bxy + cy^2 = 0\), we get \(a = 25\), \(b = -24\), and \(c = 18\).

Step 2: Use the Formula for Angle of Rotation

To eliminate the \(x'y'\)-term, use the formula \(\tan 2\theta = \frac{b}{a-c}\). Substitute the values: \(\tan 2\theta = \frac{-24}{25 - 18} = \frac{-24}{7}\).

Step 3: Calculate \(2\theta\) Using the Inverse Tangent Function

Find \(2\theta\) using the inverse tangent: \(2\theta = \arctan \left(\frac{-24}{7}\right)\). Calculate \(2\theta\) using a calculator for an approximation.

Step 4: Find \(\theta\) from \(2\theta\)

Divide \(2\theta\) by 2 to find \(\theta\). This will give the angle \(\theta\) in the first quadrant since the problem specifies that \(\theta\) is acute.

Step 5: Calculate \(\sin \theta\) and \(\cos \theta\)

Once \(\theta\) is known, calculate \(\sin \theta\) and \(\cos \theta\) directly using trigonometric functions or a calculator.

Key Concepts

Angle of Rotation

When dealing with quadratic equations such as the one in our problem, the angle of rotation is crucial for transforming the equation into a more standard form by eliminating the mixed term. This mixed term is typically represented as the coefficient of the product of two variables, in this case, the term \(x'y'\). To address this, we apply a rotation transformation that involves rotating the coordinate axes themselves.
In mathematical terms, the angle \(\theta\) of rotation can be determined using the formula \(\tan 2\theta = \frac{b}{a-c}\). This formula helps in changing the coordinate system to remove the cross-product or mixed term \(xy\). Here, \(a\), \(b\), and \(c\) represent the coefficients of \(x^2\), \(xy\), and \(y^2\) respectively. By choosing the right angle of rotation, one can simplify the equation, making it easier to work and solve with.

Key points:

• The angle of rotation helps remove the mixed term.
• Use the formula \(\tan 2\theta = \frac{b}{a-c}\) to find the needed rotation.
• This transformation makes solving the equation more manageable.

Inverse Tangent Function

The inverse tangent function, also known as \( ext{arctan}\), is central to finding the angle of rotation. Once we have the expression \(\tan 2\theta = \frac{b}{a-c}\), we're faced with determining the value of \(2\theta\). Here, the inverse tangent function comes into play to help us calculate it.
The function \(\text{arctan}(x)\) returns the angle whose tangent is \(x\). In our solution, we calculated \(2\theta\) as \(\arctan(\frac{-24}{7})\). By doing so, we're essentially "undoing" the tangent operation to revert back to the angle itself. Make sure to use a calculator for the exact value, as precision matters in subsequent calculations for trigonometric functions.
Once \(2\theta\) is obtained, divide it by 2 to find \(\theta\), since the original question asks for an acute angle. Remember, an acute angle is less than 90°, keeping \(\theta\) in a manageable range for practical applications and ease of calculation.
What's important:

• \(\text{arctan}\) helps to find \(2\theta\) from an expression involving \(\tan 2\theta\).
• The \(\text{arctan}\) function is the inverse of the tangent function.
• Precision is important; use a calculator.

Trigonometric Functions

To fully resolve the original problem, we need to dig into the world of trigonometric functions, specifically \(\sin \theta\) and \(\cos \theta\). Trigonometric functions help us understand relationships involving angles and sides in triangles, and they play a crucial role in coordinate transformations.
After determining the angle \(\theta\) using the inverse tangent function, you can compute \(\sin \theta\) and \(\cos \theta\). These functions are fundamental ratios derived from right triangles:

• \(\sin \theta\) represents the ratio of the opposite side to the hypotenuse.
• \(\cos \theta\) represents the ratio of the adjacent side to the hypotenuse.

Using a calculator, you can directly compute these trigonometric values once \(\theta\) is known. These components are essential for fully rotating the equation's variables and simplifying or solving further equations derived from the transformation process.
Highlights:

• \(\sin \theta\) and \(\cos \theta\) are derived from right triangle ratios.
• These values help further simplify the transformed equation.
• Trigonometric functions are vital for understanding angle relationships in mathematics.