Problem 90 - Contemporary Precalculus

Question

Solve the equation graphically. $$5 \sin 3 x+6 \cos 3 x=1$$

Step-by-Step Solution

Answer

Question: Solve the trigonometric equation graphically: $5 \sin 3x + 6 \cos 3x = 1$ Solution: We have expressed the given equation in terms of a single trigonometric function as: $$\sin(3x + \arctan(\frac{5}{6})) = \frac{1}{\sqrt{61}}$$ In order to solve the equation graphically, plot the functions: $$f(x) = \sin(3x + \arctan(\frac{5}{6}))$$ $$g(x) = \frac{1}{\sqrt{61}}$$ Find the points where $f(x)$ intersects $g(x)$ to obtain the solutions for $x$.

Step 1: Write the equation in the form R sin(3x + α)

We are given the equation: $$5 \sin 3x + 6 \cos 3x = 1$$ We can rewrite this equation in the form R sin(3x + α) using the angle addition formula for sine. Let R sin(α) = 5, and R cos(α) = 6. Then, the equation becomes: $$R(\sin 3x \cos \alpha + \cos 3x \sin \alpha) = 1$$ Solve for R and α: $$R = \sqrt{(5)^2 + (6)^2} = \sqrt{61}$$ $$\tan \alpha = \frac{5}{6} \Rightarrow \alpha = \arctan\left(\frac{5}{6}\right)$$ Now, the equation becomes: $$\sqrt{61} \sin(3x + \arctan(\frac{5}{6})) = 1$$

Step 2: Simplify the equation

Now, we will simplify the equation by dividing both sides by the R value: $$\sin(3x + \arctan(\frac{5}{6})) = \frac{1}{\sqrt{61}}$$

Step 3: Solve for x graphically

Now that we have expressed the equation in terms of a single trigonometric function, we can solve the equation graphically by plotting the function and finding the points where it intersects the line at the given value. To find the solution graphically, we can plot the left-hand side of the simplified equation: $$ f(x) = \sin(3x + \arctan(\frac{5}{6}))$$ And the right-hand side constant: $$g(x) = \frac{1}{\sqrt{61}}$$ The points where f(x) intersects g(x) graphically will provide the solutions for x. Remember that sine functions have a periodic nature, so you might find multiple intersections within a specified interval. If you want to find all solutions within a given range, graph the functions within that range and look for their intersections points.

Key Concepts

Graphical Solutions

In mathematics, a graphical solution is a powerful tool for understanding where two functions meet or intersect. For trigonometric equations, this method involves plotting the function and identifying where it crosses a certain value line.
Here’s how to approach it for trigonometric equations:

• Start by plotting the trigonometric function derived from the equation. In our exercise, it is given by the function \( f(x) = \sin(3x + \arctan(\frac{5}{6})) \).
• Next, plot the constant value line. In this case, \( g(x) = \frac{1}{\sqrt{61}} \).
• The solution(s) for the variable \( x \) are the \( x \)-coordinates where the graph of \( f(x) \) intersects \( g(x) \). These are the points that satisfy the original equation.

Remember, since trigonometric functions are periodic, there could be multiple intersections for the solutions. Carefully inspect the graph over the range of interest to identify all potential solutions.

Angle Addition Formula

The angle addition formula is a fundamental trigonometric identity that allows the expression of the sine or cosine of a sum of angles.
For sine, the formula is:
\[ \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \]
This formula is indispensable when converting expressions involving multiple trigonometric functions into a single one for simplification.
In our problem, we use the angle addition formula to rewrite the given equation \( 5 \sin 3x + 6 \cos 3x \) into the form \( R \sin(3x + \alpha) \). It simplifies handling equations where both sine and cosine are present.

• This was done by setting matching components of the sine and cosine terms. In this process, the values \( R \sin(\alpha) = 5 \) and \( R \cos(\alpha) = 6 \) emerge.
• Then, compute \( R = \sqrt{(5)^2 + (6)^2} = \sqrt{61} \) using the Pythagorean theorem.
• Calculate \( \alpha \) using the tangent function: \( \alpha = \arctan(\frac{5}{6}) \).

By combining these, the equation becomes manageable and ready for solving via graphical methods.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and are based on angles and their relationships with side lengths in right triangles.
The most commonly used functions are:

• Sine \( (\sin) \)
• Cosine \( (\cos) \)
• Tangent \( (\tan) \)

Each of these has distinct properties and periodicity that make them suitable for modeling cyclical and wave-like phenomena.
Here’s a brief overview:

• The sine function \( \sin(x) \) varies between -1 and 1 and repeats every \( 2\pi \) radians. It measures the ratio of the opposite side to the hypotenuse in a right triangle.
• The cosine function \( \cos(x) \) also oscillates between -1 and 1 with a period of \( 2\pi \) radians. It represents the ratio of the adjacent side to the hypotenuse.
• These functions are cyclical, meaning they repeat their pattern over an interval known as the period.

In the context of solving trigonometric equations, understanding these functions’ behaviors is essential. The sinusoidal nature allows for multiple solutions within a given range, which is why using graphical methods can visually reveal these multiple solutions easily.