Solving Cos(2x) = 11sin(x) - 5: Radians & Exact Solutions

Dec 4, 2025 by Andrew McMorgan 58 views

Solving cos(2x) = 11sin(x) - 5: A Deep Dive into Trigonometric Solutions

Hey math enthusiasts! Ever stumbled upon a trigonometric equation that seemed like a puzzle with too many pieces? Today, we're going to break down one such equation: cos(2x) = 11sin(x) - 5. Our mission? To find all the exact solutions for x within the interval [0, 2π), expressed in radians. If, after our algebraic adventure, we find no solutions exist, we’ll confidently declare “DNE” (Does Not Exist). So, buckle up, because we're about to embark on a trigonometric journey!

Transforming the Equation: A Foundation for Solutions

Let’s get started by understanding why this equation might seem a little tricky at first glance. We've got a cos(2x) term hanging out with a sin(x) term, which is like trying to compare apples and oranges. To make things easier, we need to bring them to a common ground. Thankfully, trigonometry provides us with a handy identity: the double-angle formula for cosine. We know that cos(2x) can be rewritten as 1 - 2sin²(x). This is a game-changer because now our equation will be entirely in terms of sin(x). By applying this identity, we are able to replace cos(2x) with an equivalent expression that involves only the sine function. This is a crucial step because it allows us to consolidate the terms and potentially form a more manageable equation, often a quadratic equation in disguise. This is a common strategy when dealing with trigonometric equations involving multiple angles or different trigonometric functions. By expressing everything in terms of a single function, we can utilize algebraic techniques, such as factoring or the quadratic formula, to solve for the unknown variable.

So, let’s rewrite our equation:

1 - 2sin²(x) = 11sin(x) - 5

Now, we need to rearrange the equation to set it equal to zero. This is a classic move when dealing with equations that we suspect might be quadratic in nature. By bringing all the terms to one side, we create a standard form that allows us to easily identify the coefficients and apply the appropriate solving methods.

Unveiling the Quadratic: A New Perspective

Let's move everything to one side to get a more familiar form:

0 = 2sin²(x) + 11sin(x) - 6

See that? It looks like a quadratic equation, doesn't it? We can make this even clearer by using a substitution. Let's say y = sin(x). This substitution is a powerful technique that simplifies the equation and makes it easier to work with. By temporarily replacing the trigonometric function with a single variable, we transform the equation into a more familiar algebraic form. This allows us to apply standard quadratic solving methods, such as factoring or the quadratic formula, without being distracted by the trigonometric context. Once we've solved for the temporary variable, we can then substitute back to find the values of the original trigonometric function and, ultimately, the solutions for the unknown angle.

Our equation now becomes:

2y² + 11y - 6 = 0

Now, we have a standard quadratic equation that we can solve using various methods, such as factoring or the quadratic formula. This transformation is a key step in simplifying the problem and making it accessible to standard algebraic techniques. This quadratic equation in y is much easier to handle. We can solve for y using factoring, the quadratic formula, or other methods. This is a common technique in solving trigonometric equations: transforming them into algebraic equations that we know how to solve. Once we find the values of y, we'll substitute back sin(x) for y and solve for x.

Cracking the Quadratic: Finding the Roots

Time to solve this quadratic! We can factor it:

(2y - 1)(y + 6) = 0

This gives us two possible solutions for y:

y = 1/2
y = -6

Remember, y = sin(x). So, we have:

sin(x) = 1/2
sin(x) = -6

But wait a minute! We know that the sine function has a range of [-1, 1]. That means sin(x) = -6 has no solutions. Why? Because the sine function can never output a value outside of its defined range. This is a critical understanding in trigonometry – always consider the range of trigonometric functions when solving equations. Values outside the range indicate impossible solutions. This step highlights the importance of understanding the properties and limitations of trigonometric functions. The range of sine (and cosine) is [-1, 1], so any value outside this range will not yield a valid solution.

So, we're left with:

sin(x) = 1/2

This is much more manageable! We're now down to a basic trigonometric equation. It's always a good idea to keep an eye on the possible solutions and discard any that don't make sense in the context of the trigonometric functions.

Unraveling the Sine: Finding the Angles

Now, we need to find the values of x in the interval [0, 2π) where sin(x) = 1/2. Think about the unit circle! Where is the y-coordinate (which represents sine) equal to 1/2? Visualizing the unit circle is a powerful tool for solving trigonometric equations. It helps us understand the relationships between angles and their corresponding sine and cosine values. The unit circle provides a visual representation of all possible solutions within the interval [0, 2π), making it easier to identify the angles that satisfy the given condition.

We know that sine is positive in the first and second quadrants. The reference angle for x is π/6 (30 degrees). This is a common angle that you should become familiar with in trigonometry. Recognizing these key angles and their corresponding sine and cosine values is crucial for efficient problem-solving. These angles often appear in trigonometric equations and understanding their properties can save you time and effort.

Therefore, the solutions are:

x = π/6 (in the first quadrant)
x = 5π/6 (in the second quadrant)

These are the exact solutions within our specified interval. We've successfully navigated the trigonometric terrain and pinpointed the angles that satisfy our equation.

Final Answer: The Trigonometric Triumph

So, guys, we've successfully solved the equation cos(2x) = 11sin(x) - 5 for x in radians within the interval [0, 2π). The exact solutions are:

x = π/6
x = 5π/6

We tackled this problem by using trigonometric identities, algebraic manipulation, and a good understanding of the unit circle. Remember, practice makes perfect! Keep exploring these trigonometric puzzles, and you'll become a pro in no time!

Solving trigonometric equations can be challenging, but by breaking them down into smaller steps and applying the appropriate techniques, we can find the solutions. Remember to use trigonometric identities, algebraic manipulation, and the unit circle as your tools. And always double-check your answers to make sure they make sense in the context of the equation and the trigonometric functions involved. Congrats on conquering this trigonometric challenge! You've earned your stripes as a trigonometric problem-solver. Now go forth and conquer more mathematical mysteries!