Question

Solve cos \(x \sin 2 x-2 \sin x=-2\) algebraically over the domain of real numbers. Give your answer(s) in radians.

Short answer

\(x = \frac{\pi}{2} + 2k\pi\)

Step-by-step solution

Rewrite the Equation

Rewrite the given equation to simplify: \[ \cos x \sin 2x - 2 \sin x = -2\]

Use Trigonometric Identity

Recall the trigonometric identity for \(\sin 2x\): \[ \sin 2x = 2 \sin x \cos x \]Substitute this into the equation:\[ \cos x (2 \sin x \cos x) - 2 \sin x = -2\]

Simplify the Equation

Simplify the expression:\[ 2 \cos^2 x \sin x - 2 \sin x = -2\]

Factor Out Common Terms

Factor out \(2 \sin x\) from the equation:\[ 2 \sin x (\cos^2 x - 1) = -2\]

Simplify Further

Set \(\cos^2 x - 1\) in terms of a known identity:\[ \cos^2 x - 1 = -\sin^2 x \]So, the equation simplifies to:\[ 2 \sin x (-\sin^2 x) = -2\]This simplifies further to:\[ -2 \sin^3 x = -2\]

Solve for \(\sin x\)

Divide both sides by -2:\[ \sin^3 x = 1\]Take the cube root of both sides:\[ \sin x = 1\]

Find All Solutions in Radians

We know that \(\sin x = 1\) when \(x = \frac{\pi}{2} + 2k\pi\) (where \(k\) is any integer).

Key concepts

cosine function

The cosine function, often written as \(\text{cos} x\), is one of the primary functions in trigonometry. It represents the horizontal coordinate of a point on the unit circle as an angle rotates around the origin.
The unit circle has a radius of 1, which allows the cosine of any angle to be defined as the length of the adjacent side divided by the hypotenuse in a right-angled triangle.
Several key properties of the cosine function include:

• Periodicity: The cosine function has a period of \(\text{2}\pi\) radians, which means it repeats every \(\text{2}\pi\) units.
• Range: The values of the cosine function range between \(-1\) and \(1\).
• Even Function: The cosine function is symmetric about the y-axis, meaning \(\text{cos}(-x) = \text{cos}(x)\).

In trigonometric equations, the cosine function often works alongside other functions like sine and tangent to find solutions over specific domains.

sine function

The sine function, denoted by \(\text{sin} x\), is another fundamental trigonometric function. It represents the vertical coordinate of a point on the unit circle at a given angle.
The sine function is crucial for solving various trigonometric equations:

• Periodicity: The sine function also has a period of \(\text{2}\pi\) radians.
• Range: Its values range between \(-1\) and \(1\).
• Odd Function: The sine function is an odd function, which means \(\text{sin}(-x) = -\text{sin}(x)\).

For example, in the given problem, using the identity \(\text{sin} 2x = 2 \text{sin} x \text{cos} x\) simplifies the equation. Understanding the behavior and properties of the sine function can help in recognizing patterns and suitable approaches for finding solutions.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are always true.
These identities are essential tools for simplifying and solving trigonometric equations.
Some commonly used identities include:

• Pythagorean identities: \(\text{sin}^2 x + \text{cos}^2 x = 1\)
• Double angle identities: \(\text{sin} 2x = 2 \text{sin} x \text{cos} x\)
• Sum and difference identities: \(\text{cos}(a + b) = \text{cos} a \text{cos} b - \text{sin} a \text{sin} b\)

In our example, we used the double angle identity \(\text{sin} 2x = 2 \text{sin} x \text{cos} x\) to transform and simplify the given equation. This transformation allowed us to solve the equation step by step. Without these identities, finding solutions to trigonometric equations would be much more challenging.
A good grasp of these identities is essential for efficiently tackling complex trigonometric problems.

domain

The domain of a function refers to all the possible input values (x-values) that will output a real number.
For trigonometric functions, the domain can be any set of angles.
The primary trigonometric functions \(\text{sin}\) and \(\text{cos}\) have the real number line as their domain, meaning they accept any real number as an input. In the context of the given problem, solving the equation over the domain of real numbers means finding all angles where the equation holds true.
It's important to remember that these solutions can often be represented using an infinite set of solutions, expressed using a parameter (usually denoted as k).
For example, in our case, we found the solutions in terms of radians. The key is to determine the specific x-values within the stated domain that satisfy the given equation.

radians

Radians are a unit of angular measure, where one radian is the angle created when the radius of a circle is wrapped along its circumference.
Full rotation around a circle is \(\text{2}\pi\) radians.
For practical applications:

• \frac{\text{\text{pi}}}{2}: This represents 90 degrees.
• \text{\text{pi}}: This represents 180 degrees.
• \frac{3\text{\text{pi}}}{2}: This represents 270 degrees.
• \text{2\text{\text{pi}}}: This represents 360 degrees.

Using radians simplifies many trigonometric functions and calculus operations.
In our trigonometric equation, we used radians for solutions, like \(x = \frac{\text{pi}}{2} + 2k\text{pi}\) where k is any integer. Working in radians can provide more straightforward and efficient solutions to trigonometric problems.