Solving Trigonometric Equations

1. The Core Problem: Why There's More Than One Answer

Trigonometric functions are periodic, meaning their graphs repeat in waves. If we draw a horizontal line on the graph, it will often intersect the wave multiple times. Each intersection point is a valid solution.

Your calculator will only ever give you one of these solutions, called the principal value. Our job is to use this principal value to find all the other solutions that fall within the specific range the exam question asks for (e.g., 0° ≤ x ≤ 360°).

There are two main methods to do this:

1. Using Graph Symmetries: A visual method based on sketching the trig graphs.
2. Using the Quadrant Rule (CAST diagram): A procedural method based on a diagram of the four quadrants.

Since you've mentioned finding graphs challenging but also wanting to know the most efficient methods, we will cover both. You can then choose the one you feel most confident with.

2. The Systematic Approach to Solving Trig Equations

Solving any linear trig equation follows a three-step process:

1. Isolate: Rearrange the equation so the trig function (sin(x), cos(x), or tan(x)) is the subject.
2. Principal Value: Use the inverse trig function (e.g., sin⁻¹) on your calculator to find the first angle.
3. All Solutions: Use either the graph method or the CAST method to find all other solutions in the required range.

Let's see this in action.

Method A: Using Graph Symmetry

The key is to understand the symmetry of the sin, cos, and tan graphs.

• Sine: The graph y = sin(x) is symmetrical about x = 90°. If x₁ is your principal value, the second solution is x₂ = 180° - x₁.
• Cosine: The graph y = cos(x) is symmetrical about x = 180°. If x₁ is your principal value, the second solution is x₂ = 360° - x₁.
• Tangent: The graph y = tan(x) repeats every 180°. If x₁ is your principal value, the next solution is x₂ = 180° + x₁.

The following video works through several IGCSE-style problems using this graph-sketching method. It's an excellent way to see the process applied repeatedly.

Method B: The Quadrant Rule (CAST Diagram)

This method avoids sketching graphs. It uses a diagram to determine which quadrants the solutions will be in.

The CAST diagram shows which trigonometric functions are positive in each quadrant. All in the first, Sine in the second, Tangent in the third, and Cosine in the fourth.

The steps are:

1. Find the principal value, α, using your calculator. Important: If you are solving for a negative value (e.g., sin(x) = -0.5), find the principal value for the positive equivalent first (i.e., use sin⁻¹(0.5) to get α=30°).
2. Use the CAST diagram to identify the two quadrants your solutions are in based on whether the original k was positive or negative.
3. Calculate the solutions in those quadrants using these rules:
   • Quadrant 1: x = α
   • Quadrant 2: x = 180° - α
   • Quadrant 3: x = 180° + α
   • Quadrant 4: x = 360° - α

3. Handling Negative Values and Complex Algebra

Things get slightly trickier when the equation involves more algebra or negative values. This is a common place to make mistakes.

Let's return to the Ginger Mathematician video, as he covers these cases well.

Dealing with a negative principal value from the calculator (e.g., -16.6°):

• Graph Method: Sketch the graph and mark the negative angle. Use the symmetry to find the corresponding positive angles within the 0-360° range. For sin(x) = -k, the solutions are 180° + 16.6° and 360° - 16.6°.
• CAST Method: The principal angle is α = 16.6°. Since sin(x) is negative, we need the 'T' and 'C' quadrants (3rd and 4th).
  • 3rd Quadrant: 180° + α = 180° + 16.6° = 196.6°
  • 4th Quadrant: 360° - α = 360° - 16.6° = 343.4°

4. Quadratic and Transformed Equations

Your exam will also feature more complex equation types. Here's the strategy for two common variants.

A. Quadratic Trig Equations

These look like 2cos²(x) + 3cos(x) - 2 = 0.
The strategy is summarised in the flowchart below. The key is to turn it into a standard quadratic you can solve first.

Crucial Identity: If your equation contains both sin²(x) and cos(x), you must use the identity sin²(x) + cos²(x) = 1 to get everything in terms of one function. For example, replace sin²(x) with 1 - cos²(x).

B. Equations with Transformed Angles

These look like sin(2x) = 0.5 or cos(x - 30°) = 0.8. The angle itself has been changed. This requires a specific technique to avoid missing solutions.

Let's break down step 1, as it's the most common source of error.

• If the equation is sin(2x) = 0.5 for 0° ≤ x ≤ 360°:
  • Let u = 2x.
  • The new range for u becomes 2(0°) ≤ u ≤ 2(360°), which is 0° ≤ u ≤ 720°.
• You must find all solutions for u in this much wider range (720° means two full cycles of the sine wave).
• Finally, divide all your u solutions by 2 to find the values for x.

5. Avoiding Common Exam Errors

Your goal is to build a reliable method that earns you every possible mark. Let's look at two classic traps that students fall into.

• Error 1: Dividing by a variable. In an equation like sin(x)cos(x) = sin(x), do NOT divide both sides by sin(x). You will lose the solutions where sin(x) = 0. Instead, rearrange and factorize: sin(x)cos(x) - sin(x) = 0 becomes sin(x)(cos(x) - 1) = 0. This gives two equations to solve: sin(x) = 0 and cos(x) = 1.

• Error 2: Forgetting the ± sign. When solving an equation like tan²(x) = 3, you must take both the positive and negative square roots. This means you have to solve two separate equations: tan(x) = +√3 and tan(x) = -√3, which will give you four solutions in the 0-360° range.

Conclusion

Today, we've developed a robust, systematic method for solving trigonometric equations—a high-value skill for your exam. You now have a choice of two reliable techniques (graphs or CAST) to find multiple solutions and are aware of the common traps to avoid.

Key Takeaways:

• Always follow the three-step process: Isolate, find the Principal Value, and then find All Solutions.
• You can find the other solutions using either graph sketches (visual) or the CAST diagram (procedural). Choose the one you prefer and master it.
• When your calculator gives a negative angle, use your chosen method to find the equivalent positive angles in the correct range.
• For quadratic trig equations, convert them to a standard quadratic, solve, and then solve the resulting linear trig equations.
• For transformed angles like sin(2x), always adjust the range first to ensure you don't miss solutions.
• Never divide by a trig function involving the variable; always factorize instead. Remember the ± when you take a square root.

Preview of the Next Lesson:
We are now finished with the Trigonometry module. In our next lesson, we will begin Module 11: Introduction to Calculus. We'll start with the fundamental concept of differentiation and learn how to find the gradient function of polynomial functions. This is another major topic, and we'll approach it with the same focus on clear methods and exam application.