Mathematics (0580) | Trigonometric functions

Welcome to Trigonometric Functions! (IGCSE 0580 Extended)

Hello! If you’ve already mastered using Sine, Cosine, and Tangent in right-angled triangles, you’re halfway there! This chapter takes those ratios and turns them into beautiful, continuous graphs. This is important because trigonometric functions describe natural recurring phenomena, like sound waves, tides, and pendulums. Understanding these graphs is crucial for solving more complex equations.

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1. Introduction to the Three Primary Graphs

Unlike linear or quadratic equations, trigonometric functions like sine and cosine are periodic. This means their graphs repeat the same pattern forever (or within the specified domain).

Key Terms You Need to Know:

• Period: The length of the smallest interval over which the function completes one full cycle. For \(y = \sin x\) and \(y = \cos x\), the standard period is \(360^\circ\).
• Amplitude: The maximum displacement or height of the wave from its equilibrium (centre) line. For \(y = \sin x\) and \(y = \cos x\), the standard amplitude is 1.
• Asymptote: A line that the graph approaches but never actually touches (we mostly see this in the tangent graph).

We will focus on sketching and interpreting these graphs specifically in the domain \(0^\circ \le x \le 360^\circ\).

2. The Sine Function: \(y = \sin x\)

The graph of \(y = \sin x\) is often called a sine wave or sinusoid. Think of it as a smooth, rolling ocean wave.

Key Features (for \(0^\circ \le x \le 360^\circ\)):

• Range: The y-values always stay between -1 and 1. (\(-1 \le y \le 1\))
• Amplitude: 1
• Period: \(360^\circ\)

Key Points for Sketching:

To sketch the graph accurately, you only need five key points:

1. \(x = 0^\circ\), \(y = \sin 0^\circ = 0\) (Starts at the origin)
2. \(x = 90^\circ\), \(y = \sin 90^\circ = 1\) (Hits its maximum point)
3. \(x = 180^\circ\), \(y = \sin 180^\circ = 0\) (Crosses the x-axis)
4. \(x = 270^\circ\), \(y = \sin 270^\circ = -1\) (Hits its minimum point)
5. \(x = 360^\circ\), \(y = \sin 360^\circ = 0\) (Finishes the cycle back on the x-axis)

Sketching Tip: Plot these five points and join them with a smooth, curved line (not straight segments!).

Did you know? Sine waves are used in electrical engineering to describe alternating current (AC) because the voltage flows back and forth smoothly over time.

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3. The Cosine Function: \(y = \cos x\)

The cosine graph is very closely related to the sine graph. In fact, it's exactly the same shape, just shifted!

Key Features (for \(0^\circ \le x \le 360^\circ\)):

• Range: \(-1 \le y \le 1\)
• Amplitude: 1
• Period: \(360^\circ\)

Key Points for Sketching:

The cosine graph starts high and follows the same pattern:

1. \(x = 0^\circ\), \(y = \cos 0^\circ = 1\) (Starts at its maximum point)
2. \(x = 90^\circ\), \(y = \cos 90^\circ = 0\) (Crosses the x-axis)
3. \(x = 180^\circ\), \(y = \cos 180^\circ = -1\) (Hits its minimum point)
4. \(x = 270^\circ\), \(y = \cos 270^\circ = 0\) (Crosses the x-axis again)
5. \(x = 360^\circ\), \(y = \cos 360^\circ = 1\) (Finishes the cycle back at the maximum)

Analogy: Imagine the Sine wave is a runner starting at the starting line (0,0). The Cosine wave is the same runner, but they started \(90^\circ\) (or one quarter cycle) ahead!

4. The Tangent Function: \(y = \tan x\)

The tangent graph is the "odd one out". It does not produce a smooth wave shape and has a shorter period.

Key Features (for \(0^\circ \le x \le 360^\circ\)):

• Range: The y-values are unlimited (can be anything from \(-\infty\) to \(+\infty\)).
• Period: \(180^\circ\) (The graph repeats twice in the standard \(360^\circ\) domain).
• Asymptotes: The lines where the function is undefined. These occur at \(90^\circ\) and \(270^\circ\).

Why does this happen? Remember that \(\tan x = \frac{\sin x}{\cos x}\). When \(\cos x = 0\), the fraction is undefined, which is why we get the asymptotes at \(90^\circ\) and \(270^\circ\).

Key Points for Sketching:

The graph passes through the origin and then shoots up towards the asymptote:

1. Starts: \(x = 0^\circ\), \(y = 0\)
2. Asymptote 1: Vertical line at \(x = 90^\circ\).
3. Mid-point: \(x = 180^\circ\), \(y = 0\)
4. Asymptote 2: Vertical line at \(x = 270^\circ\).
5. End: \(x = 360^\circ\), \(y = 0\)

Remember: Always draw the asymptotes as dotted lines to show they are boundaries the curve never crosses.

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5. Solving Trigonometric Equations (E7.4.2)

Solving trig equations means finding the specific angle(s) \(x\) that satisfy an equation like \(\sin x = 0.5\) within the given range (\(0^\circ \le x \le 360^\circ\)).

Don't worry if this seems tricky at first! Your calculator usually only gives you one answer (the acute one). We need a system to find all the correct answers, since the graph repeats.

The Key Tool: The CAST Diagram (or All Students Take Calculus)

The CAST diagram tells us in which quadrants (regions) of the \(360^\circ\) circle each ratio (\(\sin\), \(\cos\), \(\tan\)) is positive.

• Quadrant 1 (0 to \(90^\circ\)): All ratios are positive. Solution is the Reference Angle (\(\theta\)) itself.
• Quadrant 2 (90 to \(180^\circ\)): Sine is positive. Solution is \(180^\circ - \theta\).
• Quadrant 3 (180 to \(270^\circ\)): Tangent is positive. Solution is \(180^\circ + \theta\).
• Quadrant 4 (270 to \(360^\circ\)): Cosine is positive. Solution is \(360^\circ - \theta\).

Mnemonic: Starting in Q4 and moving anti-clockwise:
Cosine, All, Sine, Tangent (CAST)

Step-by-Step Guide to Solving \(a \sin x = k\) or \(a \cos x = k\)

Example: Solve \(2 \cos x + 1 = 0\) for \(0^\circ \le x \le 360^\circ\).

Step 1: Isolate the trigonometric function.
$$\(2 \cos x = -1\)$$ $$\(\cos x = -0.5\)$$

Step 2: Find the Reference Angle (\(\theta\)).
Ignore the negative sign for now! Use the positive value to find the acute angle (\(\theta\)).
$$\(\theta = \cos^{-1}(0.5)\)$$ $$\(\theta = 60^\circ\)$$

Step 3: Determine the relevant Quadrants.
We are solving for \(\cos x = -0.5\). Since the result is negative, we look for quadrants where Cosine is NOT positive.

• Cos is positive in A (Q1) and C (Q4).
• Therefore, Cos is negative in S (Q2) and T (Q3).

Step 4: Calculate the final solutions.
Use the reference angle (\(\theta = 60^\circ\)) and the quadrant rules:

• Solution 1 (Q2): \(x = 180^\circ - \theta = 180^\circ - 60^\circ = 120^\circ\)
• Solution 2 (Q3): \(x = 180^\circ + \theta = 180^\circ + 60^\circ = 240^\circ\)

Final Answer: \(x = 120^\circ\) or \(240^\circ\).

Common Mistake to Avoid!

When solving equations like \(\sin x = -0.7\), do not press \(\sin^{-1}(-0.7)\) on your calculator first! This will give a negative angle, which is outside your domain and harder to work with.
Always use the positive value to find the reference angle first, and then apply the CAST rules.

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6. Extended Concept: Exact Values (E7.3)

For Extended students, you must know the exact values for certain angles (\(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\)) without using a calculator. This is tested particularly in Paper 2 (Non-Calculator).

Exact Values to Memorise:

Angle (x)	\(\sin x\)	\(\cos x\)	\(\tan x\)
\(0^\circ\)	0	1	0
\(30^\circ\)	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\)
\(45^\circ\)	\(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\)	\(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\)	1
\(60^\circ\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)	\(\sqrt{3}\)
\(90^\circ\)	1	0	Undefined (Asymptote)

Memory Aid Trick: Look at the numerators for sine: \(\sqrt{0}, \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}\). Now divide them all by 2: \(\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\). Simplify, and you get the sine values! Cosine is just the reverse order.