📐 Level 2 - Topic 7: Trigonometric Equations - Solving for Angles 💪🔄

1) Introduction to Trigonometric Equations - Finding the Unknown Angles 🔄

In trigonometry, we often encounter equations that involve trigonometric functions such as sine, cosine, tangent, etc. These are called trigonometric equations. Unlike trigonometric identities, which are true for all values of the variable, trigonometric equations are true only for specific values (or sets of values) of the variable. Our goal in solving trigonometric equations is to find these specific angle values that satisfy the equation.

Trigonometric Equations vs. Identities:

Trigonometric Identities: Equations that are true for all values of the variable (for which the expression is defined). Examples: \( \sin^2(x) + \cos^2(x) = 1 \), \( \sin(2x) = 2\sin(x)\cos(x) \).
Trigonometric Equations: Equations that are true for some values of the variable. Examples: \( \sin(x) = \frac{1}{2} \), \( 2\cos(x) - 1 = 0 \).

Understanding Solutions and Periodicity:

Due to the periodic nature of trigonometric functions, trigonometric equations typically have infinitely many solutions.
We often aim to find the general solution, which represents all possible solutions, and sometimes the principal solution, which is the solution within a specific interval (usually \( [0, 2\pi) \) or \( [0^\circ, 360^\circ) \)).

Why Solve Trigonometric Equations?

Finding Angles: The primary purpose is to determine the angles that satisfy certain trigonometric relationships.
Applications: Trigonometric equations are used in physics (e.g., simple harmonic motion), engineering (e.g., signal processing), and various branches of mathematics.

In this topic, we will learn techniques to solve various types of trigonometric equations, understand how to find both principal and general solutions, and handle equations involving different trigonometric functions and algebraic forms. Let's begin our journey to find the unknown angles! 🔄📐

2) Basic Trigonometric Equations - Sine, Cosine, Tangent 🔑

Let's start with solving the most basic trigonometric equations of the form: \( \sin(x) = k \), \( \cos(x) = k \), and \( \tan(x) = k \), where \( k \) is a constant.

Definition: Principal and General Solutions for Basic Equations

For equations \( \sin(x) = k \), \( \cos(x) = k \), \( \tan(x) = k \):

Principal Solutions: These are solutions within the interval \( [0, 2\pi) \) or \( [0^\circ, 360^\circ) \). We use inverse trigonometric functions to find them.
- For \( \sin(x) = k \): Principal solutions are \( x = \arcsin(k) \) and \( x = \pi - \arcsin(k) \) (in radians) or \( x = \arcsin(k) \) and \( x = 180^\circ - \arcsin(k) \) (in degrees), provided \( -1 \leq k \leq 1 \).
- For \( \cos(x) = k \): Principal solutions are \( x = \arccos(k) \) and \( x = 2\pi - \arccos(k) \) (in radians) or \( x = \arccos(k) \) and \( x = 360^\circ - \arccos(k) \) (in degrees), provided \( -1 \leq k \leq 1 \).
- For \( \tan(x) = k \): Principal solution is \( x = \arctan(k) \) (in radians or degrees). Tangent has a period of \( \pi \) or \( 180^\circ \), so there's typically one principal solution per period we directly get from arctan.
General Solutions: To account for periodicity, we add integer multiples of the period to the principal solutions to get all solutions.
- For \( \sin(x) = k \): General solutions are \( x = \arcsin(k) + 2n\pi \) and \( x = (\pi - \arcsin(k)) + 2n\pi \) (in radians) or \( x = \arcsin(k) + n \cdot 360^\circ \) and \( x = (180^\circ - \arcsin(k)) + n \cdot 360^\circ \) (in degrees), where \( n \) is any integer.
- For \( \cos(x) = k \): General solutions are \( x = \arccos(k) + 2n\pi \) and \( x = (2\pi - \arccos(k)) + 2n\pi \) (in radians) or \( x = \arccos(k) + n \cdot 360^\circ \) and \( x = (360^\circ - \arccos(k)) + n \cdot 360^\circ \) (in degrees), where \( n \) is any integer.
- For \( \tan(x) = k \): General solution is \( x = \arctan(k) + n\pi \) (in radians) or \( x = \arctan(k) + n \cdot 180^\circ \) (in degrees), where \( n \) is any integer.

3) Equations with Multiple Angles - Adjusting for Periodicity 📐

Consider equations like \( \sin(nx) = k \), \( \cos(nx) = k \), or \( \tan(nx) = k \), where \( n \) is an integer. The presence of \( nx \) inside the trigonometric function affects the period of the function and, consequently, the number of solutions within a given interval.

Solving \( \sin(nx) = k \), \( \cos(nx) = k \), \( \tan(nx) = k \):

Find Principal Solutions for \( n x \): First, find the principal solutions for the angle \( nx \) as if it were a single variable, using inverse trigonometric functions. For example, for \( \sin(nx) = k \), find \( \alpha = \arcsin(k) \) and \( \beta = \pi - \arcsin(k) \) (principal solutions for \( \sin(\theta) = k \) are \( \theta = \alpha \) and \( \theta = \beta \)).
General Solutions for \( nx \): Write down the general solutions for \( nx \) by adding multiples of the period of the respective function. For sine and cosine, the period is \( 2\pi \), and for tangent, it's \( \pi \).
- For \( \sin(nx) = k \): \( nx = \alpha + 2m\pi \) and \( nx = \beta + 2m\pi \), where \( m \) is any integer.
- For \( \cos(nx) = k \): \( nx = \alpha + 2m\pi \) and \( nx = \beta + 2m\pi \), where \( m \) is any integer (where \( \alpha = \arccos(k), \beta = 2\pi - \arccos(k) \)).
- For \( \tan(nx) = k \): \( nx = \alpha + m\pi \), where \( m \) is any integer (where \( \alpha = \arctan(k) \)).
Solve for \( x \): Divide all parts of the general solutions by \( n \) to solve for \( x \).
- For \( \sin(nx) = k \): \( x = \frac{\alpha}{n} + \frac{2m\pi}{n} \) and \( x = \frac{\beta}{n} + \frac{2m\pi}{n} \).
- For \( \cos(nx) = k \): \( x = \frac{\alpha}{n} + \frac{2m\pi}{n} \) and \( x = \frac{\beta}{n} + \frac{2m\pi}{n} \).
- For \( \tan(nx) = k \): \( x = \frac{\alpha}{n} + \frac{m\pi}{n} \).
By substituting different integer values for \( m \), you can find as many solutions as needed, or describe the general solution set. To find principal solutions in \( [0, 2\pi) \), you can test integer values of \( m \) until the solutions fall within this range.

4) Quadratic Trigonometric Equations - Reducing to Quadratic Form 🧮

Many trigonometric equations can be transformed into quadratic equations using substitutions. These are equations that look like quadratic equations if we treat \( \sin(x) \), \( \cos(x) \), or \( \tan(x) \) as the variable. General form: \( a[\text{trig function}]^2 + b[\text{trig function}] + c = 0 \).

Solving Quadratic Trigonometric Equations:

Identify the Quadratic Form: Recognize if the equation is in the form \( a[\text{trig function}]^2 + b[\text{trig function}] + c = 0 \), where the "trig function" could be \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), etc.
Substitution: Let \( y = [\text{trig function}] \). Substitute \( y \) into the trigonometric equation to get a standard quadratic equation in \( y \): \( ay^2 + by + c = 0 \).
Solve the Quadratic Equation for \( y \): Use factoring, quadratic formula, or other methods to solve for \( y \). You will get one or two values for \( y \).
Back-Substitution and Solve for \( x \): Substitute back \( [\text{trig function}] \) for \( y \) and solve the resulting basic trigonometric equations for \( x \). For each value of \( y \) you found, you will have an equation like \( \sin(x) = y_1 \), \( \cos(x) = y_2 \), etc. Solve these to find the values of \( x \). Remember to find general solutions if needed, considering periodicity.
Check for Validity: Ensure that the values of \( y \) you found are within the valid range for the trigonometric function (e.g., \( -1 \leq \sin(x) \leq 1 \) and \( -1 \leq \cos(x) \leq 1 \)). If any value of \( y \) is outside the valid range, it will not yield real solutions for \( x \).

5) Equations with Multiple Trigonometric Functions - Simplification Techniques 🛠️

Solving equations that involve multiple trigonometric functions (e.g., mixing sine, cosine, tangent) often requires using trigonometric identities to simplify the equation into a solvable form. Here are common techniques:

Simplification Techniques:

Use Trigonometric Identities: Apply identities (Pythagorean, quotient, reciprocal, etc.) to express all trigonometric functions in terms of a single trigonometric function (e.g., all in terms of sine or cosine) or to simplify expressions. For example, use \( \sin^2(x) + \cos^2(x) = 1 \) to replace \( \cos^2(x) \) with \( 1 - \sin^2(x) \) or vice-versa. Or use \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
Factorization: After simplification, try to factor the equation. Setting each factor to zero can lead to simpler equations to solve.
Squaring Both Sides: If necessary, square both sides of the equation to eliminate square roots or to utilize Pythagorean identities. Caution: Squaring can introduce extraneous solutions (solutions that do not satisfy the original equation). Always check your solutions in the original equation when you square both sides.
Convert to a Common Function: Try to rewrite the equation in terms of a single trigonometric function and a single angle variable if possible.
Isolate Trigonometric Function: Aim to isolate a single trigonometric function on one side of the equation to get it in the form like \( \sin(x) = k \), \( \cos(x) = k \), \( \tan(x) = k \), which we know how to solve.

6) Examples - Solving Trigonometric Equations 🚀

Example 1: Basic Sine Equation

Solve \( \sin(x) = \frac{1}{2} \) for general solutions in radians and principal solutions in \( [0, 2\pi) \).

Principal value: \( \arcsin(\frac{1}{2}) = \frac{\pi}{6} \).
Principal solutions: \( x_1 = \frac{\pi}{6} \), \( x_2 = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \). Both are in \( [0, 2\pi) \).
General solutions: \( x = \frac{\pi}{6} + 2n\pi \) and \( x = \frac{5\pi}{6} + 2n\pi \), where \( n \) is any integer.

Solution: Principal solutions are \( x = \frac{\pi}{6}, \frac{5\pi}{6} \). General solutions are \( x = \frac{\pi}{6} + 2n\pi, \frac{5\pi}{6} + 2n\pi, n \in \mathbb{Z} \).

Example 2: Equation with Multiple Angle

Solve \( \cos(2x) = -\frac{\sqrt{3}}{2} \) for principal solutions in \( [0, 2\pi) \).

Principal value for \( 2x \): \( \arccos(-\frac{\sqrt{3}}{2}) = \frac{5\pi}{6} \).
Other principal solution for \( 2x \): \( 2\pi - \frac{5\pi}{6} = \frac{7\pi}{6} \).
General solutions for \( 2x \): \( 2x = \frac{5\pi}{6} + 2n\pi \) and \( 2x = \frac{7\pi}{6} + 2n\pi \).
Solve for \( x \): \( x = \frac{5\pi}{12} + n\pi \) and \( x = \frac{7\pi}{12} + n\pi \).
Principal solutions for \( x \) in \( [0, 2\pi) \): For \( x = \frac{5\pi}{12} + n\pi \): For \( n=0, x = \frac{5\pi}{12} \). For \( n=1, x = \frac{5\pi}{12} + \pi = \frac{17\pi}{12} \). For \( n=2, x = \frac{5\pi}{12} + 2\pi > 2\pi \). For \( n=-1, x < 0 \). For \( x = \frac{7\pi}{12} + n\pi \): For \( n=0, x = \frac{7\pi}{12} \). For \( n=1, x = \frac{7\pi}{12} + \pi = \frac{19\pi}{12} \). For \( n=2, x = \frac{7\pi}{12} + 2\pi > 2\pi \). For \( n=-1, x < 0 \). Principal solutions: \( x = \frac{5\pi}{12}, \frac{7\pi}{12}, \frac{17\pi}{12}, \frac{19\pi}{12} \).

Solution: Principal solutions are \( x = \frac{5\pi}{12}, \frac{7\pi}{12}, \frac{17\pi}{12}, \frac{19\pi}{12} \).

Example 3: Quadratic Trigonometric Equation

Solve \( 2\sin^2(x) - \sin(x) - 1 = 0 \) for principal solutions in \( [0, 2\pi) \).

Substitution: Let \( y = \sin(x) \). Equation becomes \( 2y^2 - y - 1 = 0 \).
Solve quadratic: Factor or use quadratic formula. \( (2y + 1)(y - 1) = 0 \). So, \( y = 1 \) or \( y = -\frac{1}{2} \).
Back-substitute: \( \sin(x) = 1 \) or \( \sin(x) = -\frac{1}{2} \).
Solve \( \sin(x) = 1 \): Principal solution \( x = \frac{\pi}{2} \).
Solve \( \sin(x) = -\frac{1}{2} \): Reference angle \( \arcsin(\frac{1}{2}) = \frac{\pi}{6} \). Sine is negative in quadrants III and IV. Principal solutions are \( x = \pi + \frac{\pi}{6} = \frac{7\pi}{6} \) and \( x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \).

Solution: Principal solutions are \( x = \frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6} \).

Example 4: Equation with Multiple Functions

Solve \( \sin(x)\cos(x) = \cos(x) \) for principal solutions in \( [0, 2\pi) \).

Rearrange and Factor: \( \sin(x)\cos(x) - \cos(x) = 0 \Rightarrow \cos(x)(\sin(x) - 1) = 0 \).
Set each factor to zero: \( \cos(x) = 0 \) or \( \sin(x) - 1 = 0 \Rightarrow \sin(x) = 1 \).
Solve \( \cos(x) = 0 \): Principal solutions \( x = \frac{\pi}{2}, \frac{3\pi}{2} \).
Solve \( \sin(x) = 1 \): Principal solution \( x = \frac{\pi}{2} \). (Already listed).

Solution: Principal solutions are \( x = \frac{\pi}{2}, \frac{3\pi}{2} \).

7) Practice Questions 🎯

7.1 Fundamental – Solving Basic Trigonometric Equations

1. Solve \( \sin(x) = \frac{\sqrt{3}}{2} \) for general solutions in radians and principal solutions in \( [0, 2\pi) \).

2. Solve \( \cos(x) = -\frac{1}{2} \) for general solutions in degrees and principal solutions in \( [0^\circ, 360^\circ) \).

3. Solve \( \tan(x) = 1 \) for general solutions in radians and principal solutions in \( [0, \pi) \).

4. Find the principal solutions of \( 2\sin(x) - 1 = 0 \) in \( [0, 2\pi) \).

5. Find the principal solutions of \( \sqrt{2}\cos(x) + 1 = 0 \) in \( [0^\circ, 360^\circ) \).

6. Solve \( \tan(x) + \sqrt{3} = 0 \) for general solutions in radians.

7. Find the principal solutions of \( \sin(2x) = 1 \) in \( [0, 2\pi) \).

8. Find the principal solutions of \( \cos(3x) = \frac{1}{2} \) in \( [0^\circ, 360^\circ) \).

9. Solve \( 2\cos^2(x) - 3\cos(x) + 1 = 0 \) for principal solutions in \( [0, 2\pi) \).

10. Solve \( \tan^2(x) - 3 = 0 \) for general solutions in degrees.

11. Find the principal solutions for \( \sin(x)\cos(x) = 0 \) in \( [0, 2\pi) \).

12. Solve \( \sin(x) = \sin(2x) \) for principal solutions in \( [0, 2\pi) \). (Hint: Use double angle identity).

7.2 Challenging – Advanced Equations and Problem Solving 💪🚀

1. Solve \( \sin(x) + \cos(x) = 1 \) for principal solutions in \( [0, 2\pi) \). (Hint: Square both sides, be careful of extraneous solutions).

2. Solve \( \sin(2x) = \cos(x) \) for principal solutions in \( [0^\circ, 360^\circ) \). (Hint: Double angle identity).

3. Find the general solution of \( \tan(x) + \cot(x) = 2 \). (Hint: Express cot in terms of tan).

4. Solve \( \sin^2(x) - 2\sin(x)\cos(x) - 3\cos^2(x) = 0 \) for general solutions in radians. (Hint: Divide by \( \cos^2(x) \) assuming \( \cos(x) \neq 0 \)).

5. (Conceptual) Discuss the number of solutions for \( \sin(nx) = k \) in the interval \( [0, 2\pi) \) as \( n \) varies (where \( n \) is a positive integer and \( -1 < k < 1 \)). How does the value of \( n \) affect the number of principal solutions?

8) Summary - Mastering Trigonometric Equations 🎉

Trigonometric Equations vs. Identities: Equations solved for specific angles, not true for all angles.
Basic Equations: Solving \( \sin(x) = k \), \( \cos(x) = k \), \( \tan(x) = k \) using inverse functions.
Principal Solutions: Solutions in \( [0, 2\pi) \) or \( [0^\circ, 360^\circ) \).
General Solutions: Account for periodicity, using \( + 2n\pi \) or \( + n\pi \).
Equations with Multiple Angles: Solve for \( nx \) first, then for \( x \), adjust for period change.
Quadratic Equations: Use substitution to reduce to quadratic form.
Multiple Functions: Simplify using identities, factor, square (with care), and aim for single function equations.
Always check solutions, especially after squaring.

Excellent work! You've now navigated the world of trigonometric equations and learned various techniques to solve them. Understanding how to find both principal and general solutions is crucial for many applications of trigonometry. As you move forward, you'll find these skills invaluable in more advanced mathematics and sciences. Keep practicing and you'll become adept at unlocking the angles hidden within trigonometric equations! 🔄📐💪🌟