Problem 71 - Calculus for Biology and Medicine

Question

Solve \(\cos ^{2} \theta-2=2 \sin \theta\) on \([0,2 \pi)\).

Step-by-Step Solution

Answer

The solution is \( \theta = \frac{\pi}{2} \).

Step 1: Express Cosine in terms of Sine

Recall that the Pythagorean identity states that \( \cos^2 \theta = 1 - \sin^2 \theta \). Substitute \( \cos^2 \theta \) in the equation: \( (1 - \sin^2 \theta) - 2 = 2 \sin \theta \).

Step 2: Simplify the Equation

Simplify the equation from Step 1 to get \( 1 - \sin^2 \theta - 2 = 2 \sin \theta \). This simplifies to \( -\sin^2 \theta + 2 \sin \theta - 1 = 0 \).

Step 3: Rearrange to a Quadratic Equation

Multiply through by -1 to have \( \sin^2 \theta - 2 \sin \theta + 1 = 0 \). This is a quadratic equation in terms of \( \sin \theta \).

Step 4: Factor the Quadratic Equation

Notice that the quadratic equation \( \sin^2 \theta - 2 \sin \theta + 1 = 0 \) factors as \( (\sin \theta - 1)^2 = 0 \).

Step 5: Solve for Sine Values

From \((\sin \theta - 1)^2 = 0\), we have \( \sin \theta = 1 \). We need to find \( \theta \) such that \( \sin \theta = 1 \).

Step 6: Find the Values of \( \theta \)

On the interval \([0, 2\pi)\), \( \sin \theta = 1 \) occurs only at \( \theta = \frac{\pi}{2} \).

Step 7: Verify the Solution

Verify that substituting \( \theta = \frac{\pi}{2} \) back into the original equation satisfies it: \( \cos^2(\frac{\pi}{2}) - 2 = 2 \sin(\frac{\pi}{2}) \) simplifies to \( 0 - 2 = 2 \), which is true, confirming the solution.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, akin to the renowned Pythagorean theorem. In the world of trigonometry, it connects the squares of the sine and cosine of the same angle. Specifically, it states that for any angle \( \theta \), the identity is given by:
\[\cos^2 \theta + \sin^2 \theta = 1\]
This identity is incredibly useful because it allows us to express one trigonometric function in terms of another.
For instance, if you rearrange it, you can express cosine in terms of sine:

• \( \cos^2 \theta = 1 - \sin^2 \theta \)
• Or, vice versa, \( \sin^2 \theta = 1 - \cos^2 \theta \)

The main advantage of using this identity lies in simplifying trigonometric equations. By swapping cosine or sine with the other, we can make complex equations easier to solve. Keeping this identity in mind allows us to tackle problems in both algebraic and geometric contexts effectively.

Sine Function

The sine function is one of the primary trigonometric functions. It is essential in understanding periodic phenomena, such as waves. Given an angle \( \theta \), the sine function, denoted as \( \sin \theta \), represents the y-coordinate of a point on the unit circle.
The unit circle is a circle of radius one, centered at the origin. Increasing or decreasing \( \theta \) moves the point around the circle, creating a wave-like pattern when plotted over time:

• The range of sine is \([-1, 1]\), as no y-coordinate on the unit circle falls outside this range.
• It has a period of \(2\pi\), meaning if \( \theta \) increases by \(2\pi\) the sine value repeats.

In trigonometric equations, \( \sin \theta = a \) is equivalent to identifying angles that give a certain height on the unit circle. This property is particularly relevant when solving equations involving \( \sin^2 \theta \), as substituting \( \sin \theta \) brings the equation into a solvable form.
Understanding the sine function's properties helps in efficiently solving trigonometric problems, finding patterns, and verifying solutions.

Quadratic Equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where:

• \( a, b, \) and \( c \) are constants,
• \( a eq 0 \).

The quadratic equation in inherent trigonometric problems, especially those with squared terms in the sine or cosine functions. For example, in our exercise, after applying identities, we arrived at:\[\sin^2 \theta - 2 \sin \theta + 1 = 0\]This equation resembles the standard quadratic form with \( \sin \theta \) playing the role of the variable, \( x \).
Factoring is one of the primary methods used to solve quadratic equations:

• It involves finding values such that the expression is a product of two binomial expressions.
• When a factor is repeated, as in our example, \((\sin \theta - 1)^2 = 0\), this means the solution is a repeated root.
• The solution to this quadratic equation indicates that \( \sin \theta = 1 \).

Solving quadratic equations in trigonometry often requires transforming the equation into a manageable form using trigonometric identities. Recognizing these patterns helps make the problem-solving process clearer and more straightforward.