Problem 39 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \sec x \sin ^{2} x=\tan x $$

Step-by-Step Solution

Answer

Solutions: \( x = n \cdot 180 \) or \( x = 90 + m \cdot 360 \), \( n, m \in \mathbb{Z} \).

Step 1: Express Trigonometric Functions

Express all trigonometric identities in terms of sine and cosine to simplify the equation. We know that \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Substitute these into the equation: \( \frac{1}{\cos x}\sin^2 x = \frac{\sin x}{\cos x} \).

Step 2: Simplify the Equation

Multiply both sides by \( \cos x \) to eliminate the denominator: \( \sin^2 x = \sin x \).

Step 3: Factor the Equation

Factor \( \sin x \) from the left side of the equation: \( \sin x(\sin x - 1) = 0 \).

Step 4: Determine Solutions of Factored Equation

Set each factor equal to zero: \( \sin x = 0 \) or \( \sin x - 1 = 0 \).

Step 5: Solve \( \sin x = 0 \)

The equation \( \sin x = 0 \) means that \( x = n\pi \), where \( n \) is an integer. In degrees, this corresponds to \( x = 180n \).

Step 6: Solve \( \sin x = 1 \)

The equation \( \sin x = 1 \) implies that \( x = \frac{\pi}{2} + 2m\pi \), where \( m \) is an integer. In degrees, this corresponds to \( x = 90 + 360m \).

Step 7: Combine Solutions

Thus, the solutions are \( x = n \cdot 180 \) or \( x = 90 + m \cdot 360 \), where \( n \) and \( m \) are integers. In the range of degrees, these solutions include values like \( 0, 90, 180, 270, 360, 450, \) etc.

Key Concepts

Trigonometric Identities

In trigonometry, identities are equations that hold true for all values of the variable. They serve as tools to simplify trigonometric expressions and solve equations. The two primary identities used in the given trigonometric equation are related to secant and tangent. - The identity for secant (\[ \sec x = \frac{1}{\cos x} \]) allows transforming secant functions into terms of cosine. - The identity for tangent (\[ \tan x = \frac{\sin x}{\cos x} \]) converts tangent functions into sine and cosine.This step of using identities helps in reducing the complexity of the equation, making it easier to solve. Recognizing and applying these identities is crucial in solving trigonometric equations efficiently.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions which define the relationship between the angles and sides of right triangles. In terms of the unit circle, - Sine (\[ \sin x \]) represents the y-coordinate, which describes the vertical position on the circle.- Cosine (\[ \cos x \]) provides the x-coordinate, indicating the horizontal position.Understanding these functions in terms of the unit circle helps in interpreting solutions over different angles. Simplifying expressions to include only these functions can effectively tackle many trigonometric problems. In our exercise, converting secant and tangent to sine and cosine significantly reduced the problem to a much simpler form.

Solving Equations

Solving trigonometric equations involves finding all angles that satisfy the equation. The process may include:- Simplifying the expression using identities.- Eliminating denominators by multiplying or dividing across the equation.- Factoring, which is breaking down the equation into the product of simpler expressions.After factoring, setting each component factor to zero helps us find the potential solutions. For example, when \( \sin x(\sin x - 1) = 0 \), it gives two simpler equations: \( \sin x = 0 \) and \( \sin x = 1 \), each representing a subset of solutions. Solving these allows us to determine specific angles where the original equation holds true.

Angle Measurement in Degrees

Angles in trigonometric problems can be measured in either degrees or radians. Here, we're utilizing degrees, which is a more intuitive measurement for many learners. - Degrees divide the circle into 360 parts, where a full rotation equals 360°. - Familiar angles include 0°, 90°, 180°, and so on.When solving the provided trigonometric equation, expressing the solutions in degrees makes it easier to visualize and comprehend. For example, solutions like 0, 90, 180 give clear, direct positions on the unit circle, connecting the abstract mathematics to real-world understanding. Expressing these solutions in terms like \( x = 180n \) and \( x = 90 + 360m \) clarifies the infinite periodic nature of solutions and fits trigonometric problems contextually in degree measures.