Introduction to Trigonometric Equations
Welcome to one of the most practical parts of your A Level journey! In this chapter, we are going to learn how to solve trigonometric equations. Think of these like the "inverse" of what you’ve done before. Instead of being told an angle and finding a value, you're given a value (like \(0.5\)) and you need to find all the possible angles that produce it.
Why does this matter? Trigonometry isn't just about triangles; it's the language of waves. Whether you are studying sound waves in music, light waves in physics, or the tides in geography, you are using the math we are about to cover!
1. Solving Simple Equations
The simplest trig equations look like this: \(\sin \theta = 0.5\) or \(\cos x = -\frac{1}{\sqrt{2}}\). Your goal is to find all values of the angle within a specific interval (usually \(0^\circ \leq \theta \leq 360^\circ\) or \(0 \leq \theta \leq 2\pi\)).
The Principal Value
When you type \(\sin^{-1}(0.5)\) into your calculator, it gives you \(30^\circ\). This is called the Principal Value. However, because trig graphs repeat themselves (they are periodic), there are usually other angles that work too!
Finding Other Solutions
Don't worry if this seems tricky at first—most students find the second solution the hardest part. You can find extra solutions using two main methods:
- The Graph Method: Look at the symmetry of the sine, cosine, or tangent graphs.
- The CAST Diagram: A circle divided into four quadrants showing where each function is positive (All, Sine, Tan, Cosine).
Quick Review - The "Second Solution" Rules:
For \(0^\circ \leq \theta \leq 360^\circ\):
- For Sine: \(180^\circ - \text{Principal Value}\)
- For Cosine: \(360^\circ - \text{Principal Value}\)
- For Tangent: \(180^\circ + \text{Principal Value}\) (then keep adding/subtracting \(180^\circ\))
Common Mistake: Always check your calculator is in the right mode! If the question uses degrees (\(^\circ\)), use Degree mode. If it uses \(\pi\), use Radian mode.
Summary Takeaway: Your calculator only tells you half the story. Always use the symmetry of the graph or a CAST diagram to find the other solutions in the given range.
2. Using Identities to Simplify
Sometimes an equation has both \(\sin\) and \(\cos\) in it. To solve these, we need to use our trigonometric identities to turn everything into one type of trig function.
Identity 1: Turning Sin and Cos into Tan
If you see an equation like \(3\sin \theta = 2\cos \theta\), you can divide both sides by \(\cos \theta\).
Since \(\frac{\sin \theta}{\cos \theta} = \tan \theta\), the equation becomes \(3\tan \theta = 2\), or \(\tan \theta = \frac{2}{3}\). Now it’s easy to solve!
Identity 2: The Pythagorean Identity
The most famous identity is: \(\sin^2 \theta + \cos^2 \theta = 1\).
This is incredibly useful when you have a mix of \(\sin^2\) and \(\cos\). For example, if you see a \(\cos^2 \theta\), you can swap it for \((1 - \sin^2 \theta)\). This helps you get every term into the same function.
Did you know? This identity is actually just Pythagoras' Theorem (\(a^2 + b^2 = c^2\)) hidden inside a circle with a radius of 1!
Summary Takeaway: If an equation has two different trig functions, use identities to rewrite it so it only uses one (usually by turning everything into \(\tan \theta\) or using \(\sin^2 \theta + \cos^2 \theta = 1\)).
3. Quadratic Trigonometric Equations
Sometimes trig equations look like quadratic equations. For example:
\(2\cos^2 \theta - \cos \theta - 1 = 0\)
Step-by-Step Process:
- Substitution: To make it look less scary, let \(y = \cos \theta\). Now you have \(2y^2 - y - 1 = 0\).
- Factorise: Solve it like a normal quadratic: \((2y + 1)(y - 1) = 0\).
- Solve for y: You get \(y = -0.5\) and \(y = 1\).
- Solve for \(\theta\): Replace \(y\) with \(\cos \theta\), so solve \(\cos \theta = -0.5\) and \(\cos \theta = 1\) separately.
Common Mistake: Forgetting that \(\sin \theta\) and \(\cos \theta\) must be between \(-1\) and \(1\). If your quadratic gives you \(\sin \theta = 2\), just write "no solution" for that part and move on!
Summary Takeaway: Treat \(\sin \theta\) or \(\cos \theta\) as a single variable (like \(x\)). Solve the quadratic first, then find the angles for each result.
4. Equations with Multiple Angles (\(2\theta\), \(3\theta\), etc.)
What if the equation is \(\sin(2\theta) = 0.5\)? This means the wave is moving twice as fast, so you’ll likely have more solutions.
The "Adjusted Range" Trick
If the interval is \(0^\circ \leq \theta \leq 360^\circ\), but the angle is \(2\theta\), you must look for solutions in the range \(0^\circ \leq 2\theta \leq 720^\circ\).
- Step 1: Find all values for \(2\theta\) up to \(720^\circ\).
- Step 2: Then divide every answer by 2 to get the values for \(\theta\).
Analogy: Imagine \(\theta\) is a car and \(2\theta\) is a car going double the speed. In the same amount of time, the faster car will complete twice as many laps of the track (the circle), meaning more solutions!
Summary Takeaway: Always find all the solutions for the multiple angle first, then divide at the very last step. Never divide the value inside the bracket at the start!
5. Using Compound and Double Angle Formulae
In your MEI H640 course, you are also expected to use compound angle formulae to solve more complex equations.
The Double Angle Formulae
These are essential for your exams:
- \(\sin 2\theta = 2\sin \theta \cos \theta\)
- \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta\)
Memory Aid: For \(\cos 2\theta\), choose the version that matches the other terms in your equation. If the rest of the equation is in \(\sin\), use \(1 - 2\sin^2 \theta\).
The Form \(a\cos \theta + b\sin \theta\)
If you see \(\sin\) and \(\cos\) added together (like \(3\cos \theta + 4\sin \theta = 2\)), use the R-addition method to turn them into a single wave:
\(R\cos(\theta - \alpha)\) or \(R\sin(\theta + \alpha)\)
Where \(R = \sqrt{a^2 + b^2}\) and \(\alpha\) is found using \(\tan \alpha = \frac{b}{a}\).
Summary Takeaway: Double angle formulae help break down \(2\theta\) into single \(\theta\) terms. The \(R\cos(\theta \pm \alpha)\) method is your go-to tool for solving linear combinations of \(\sin\) and \(\cos\).