Problem 1 - Precalculus Mathematics for Calculus

Question

We can use identities to help us solve trigonometric equations. Using a Pythagorean identity we see that the equation \(\sin x+\sin ^{2} x+\cos ^{2} x=1\) is equivalent to the basic equation _____ whose solutions are \(x=\) _____.

Step-by-Step Solution

Answer

\(\sin x = 0\); \(x = n\pi\) for integers \(n\).

Step 1: Understanding the Given Equation

We start with the equation \( \sin x + \sin^2 x + \cos^2 x = 1 \). The task is to simplify it using trigonometric identities.

Step 2: Utilize the Pythagorean Identity

Use the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \). Substitute this identity into the equation: \( \sin^2 x + \cos^2 x = 1 \). This simplifies the equation to \( \sin x + 1 = 1 \).

Step 3: Simplification

After the substitution, we have \( \sin x + 1 = 1 \). Subtracting 1 from both sides, we obtain \( \sin x = 0 \).

Step 4: Solve for \(x\)

The equation \( \sin x = 0 \) has solutions where \( x = n\pi \) for any integer \( n \). This is because the sine function equals zero at integer multiples of \( \pi \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry involving the sine and cosine functions. It states that for any angle \( x \), the sum of the squares of the sine and cosine functions equals one:

• \( \sin^2 x + \cos^2 x = 1 \).

This identity is crucial because it allows us to connect and transform different parts of trigonometric equations. It's named after the Pythagorean theorem since it mimics the relationship between the sides of a right triangle. In many trigonometric problems, like the one we encountered, using this identity helps to simplify and solve equations effectively.
In our problem, the use of the Pythagorean identity allowed us to convert \( \sin^2 x + \cos^2 x = 1 \) into a simpler form that immediately led us to a solution. Once realized, it streamlined the equation to identify that \( \sin x = 0 \). This use transforms seemingly complex trigonometric equations into forms that we can easily tackle.

Sine Function

The sine function is a trigonometric function that relates to an angle in the context of a right triangle. It measures the ratio of the length of the side opposite the angle to the hypotenuse. In the unit circle, \( \sin x \) represents the y-coordinate of a point on the circle.

• The function is periodic with a period of \( 2\pi \).
• It has a range of \([-1, 1]\).

This periodic nature means that the sine function repeats its values in regular intervals. For instance, if \( \sin x = 0 \), then x can be any integer multiple of \( \pi \) (i.e., \( x = n\pi \) where \( n \) is an integer), reflecting the points where the curve intersects the x-axis.
In the specific problem, recognizing that \( \sin x = 0 \) helps us determine all possible solutions for \( x \). Knowing these basic properties of the sine function can dramatically simplify solving equations that involve it.

Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent, and require certain skills and strategies to solve. Solving these equations often involves simplifying the equation using known identities, like the Pythagorean identity, and then finding the general solutions or specific values that satisfy the equation.

• Strategies include using identities to simplify expressions.
• Knowing the periodic properties of functions.
• Determining solutions like multiples of \( \pi \) when dealing with the sine and cosine functions.

In the example we solved, simplifying the given equation using the Pythagorean identity reduced it to the well-known equation \( \sin x = 0 \). Solving this equation involves recognizing the pattern of the sine function's zeros, which naturally occur at integer multiples of \( \pi \). Understanding that trigonometric functions are cyclical helps us list out all possible solutions, accommodating the periodicity of these functions across all angles.
Mastering these techniques in handling trigonometric equations is essential for tackling both elementary problems and more challenging trigonometric applications in mathematics.