Equations

Welcome to Trigonometric Equations!

In this chapter, we are going to learn how to solve equations involving sine, cosine, and tangent. If you’ve ever looked at a wave or a swinging pendulum, you’ve seen trigonometry in action. Solving these equations is simply the process of finding the specific angles where these "waves" hit a certain value.

Don't worry if this seems a bit "loopy" at first—trigonometry is periodic (it repeats!), so once you find the pattern, you'll find the answers!

1. The Basics: What is an Inverse?

When we have an equation like \(\sin \theta = 0.5\), we want to find the angle \(\theta\). To do this, we use the inverse functions on our calculator: arcsin (\(\sin^{-1}\)), arccos (\(\cos^{-1}\)), and arctan (\(\tan^{-1}\)).

The Principal Value

When you type \(\sin^{-1}(0.5)\) into your calculator, it gives you \(30^{\circ}\). This is called the Principal Value. It is the "main" answer, but because trigonometric graphs repeat forever, it is usually not the only answer.

Analogy: Imagine a Ferris wheel. If I ask, "At what time is the carriage 10 meters high?", there might be two times during one full circle (once going up, once going down). Your calculator only tells you the first one!

Quick Review:
- Inverse Sine: \(\arcsin x\) or \(\sin^{-1} x\)
- Inverse Cosine: \(\arccos x\) or \(\cos^{-1} x\)
- Inverse Tangent: \(\arctan x\) or \(\tan^{-1} x\)

2. Finding All the Solutions

Most exam questions will ask you to find all solutions within a specific range, usually \(0^{\circ} \leq \theta \leq 360^{\circ}\). To find the "hidden" second solutions, we use the CAST diagram or the symmetry of the graphs.

The CAST Diagram Mnemonic

The CAST diagram tells us which trig functions are positive in each quadrant (starting from the top right and moving counter-clockwise):
- Q1 (0-90°): All are positive.
- Q2 (90-180°): Sine is positive.
- Q3 (180-270°): Tan is positive.
- Q4 (270-360°): Cosine is positive.

Memory Aid: Add Sugar To Coffee or All Students Take Calculus.

Step-by-Step: Solving \(\sin \theta = k\)

1. Use your calculator to find the Principal Value: \(\theta = \sin^{-1}(k)\).
2. Find the second solution for Sine: \(180^{\circ} - \text{Principal Value}\).
3. For Cosine: \(360^{\circ} - \text{Principal Value}\).
4. For Tangent: \(\text{Principal Value} + 180^{\circ}\).

Summary Takeaway: Always check your range! If your angle repeats every \(360^{\circ}\) (for sin and cos) or every \(180^{\circ}\) (for tan), you might need to add or subtract these values to stay inside the limits given in the question.

3. Using Identities to Solve Equations

Sometimes an equation has both sine and cosine, which makes it hard to solve. We use identities to turn it into an equation with just one trig function.

Identity 1: The Tangent Relationship

\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]

Example: Solve \(\sin \theta = 3 \cos \theta\).
Divide both sides by \(\cos \theta\):
\(\frac{\sin \theta}{\cos \theta} = 3\)
\(\tan \theta = 3\)
Now you can just use \(\tan^{-1}(3)\) to find your angles!

Identity 2: The Pythagoras Connection

\[\sin^2 \theta + \cos^2 \theta = 1\]

This is incredibly useful when you have a "quadratic-style" equation, like one containing both \(\sin^2 \theta\) and \(\cos \theta\). You can swap \(\sin^2 \theta\) for \((1 - \cos^2 \theta)\).

Common Mistake to Avoid: Never divide an equation by \(\sin \theta\) or \(\cos \theta\) if it's the only way to get rid of them. You might "cancel out" a valid solution! Always try to factorize instead.

4. Working with Multiple Angles (e.g., \(2\theta\))

If you see an equation like \(\sin(2\theta) = 0.5\), don't panic! It just means the wave is moving twice as fast.

The "Change the Range" Method

1. Adjust the range: If the range for \(\theta\) is \(0 \leq \theta \leq 360\), then the range for \(2\theta\) is \(0 \leq 2\theta \leq 720\).
2. Solve for the whole bracket: Let \(X = 2\theta\). Solve \(\sin X = 0.5\) for all values of \(X\) up to \(720^{\circ}\).
3. Divide at the end: Once you have all your values for \(X\), divide them all by \(2\) to find your final values for \(\theta\).

Did you know? In music, if you double the frequency of a sound wave (like changing \(\theta\) to \(2\theta\)), the pitch goes up exactly one octave!

5. Quadratic Trigonometric Equations

Some equations look like algebra problems: \(2\cos^2 \theta - \cos \theta - 1 = 0\).

The Trick: Treat \(\cos \theta\) like a normal \(x\).
Let \(x = \cos \theta\).
The equation becomes \(2x^2 - x - 1 = 0\).
Factorize it: \((2x + 1)(x - 1) = 0\).
This gives you two simple equations to solve: \(\cos \theta = -0.5\) and \(\cos \theta = 1\).

Summary Takeaway: If you see a trig function squared (\(\sin^2\), \(\cos^2\)), think "Quadratic"! Use substitution to make it look less scary.

Final Quick Review Box

Key Points for the Exam:
- Check your units: Is the calculator in Degrees or Radians? (H630 AS Level focuses on Degrees).
- Principal Value: The first answer from your calculator.
- Secondary Values: Use \(180 - \theta\) for Sine, \(360 - \theta\) for Cosine, and \(180 + \theta\) for Tangent.
- Identities: Use \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sin^2 \theta + \cos^2 \theta = 1\) to simplify.
- Intervals: Always check if your answers fit between the boundaries given (e.g., \(0\) to \(360\)).