Trigonometry

Q: How do you convert between degrees and radians?

To convert degrees to radians, multiply by $\frac{\pi}{180}$; to convert radians to degrees, multiply by $\frac{180}{\pi}$.

Q: What is the formula for the arc length of a circle sector?

The arc length $s$ is given by $s = r\theta$, where $r$ is the radius and $\theta$ is the angle in radians.

Q: What is the formula for the area of a circle sector?

The area $A$ is given by $A = \frac{1}{2}r^2\theta$, where $r$ is the radius and $\theta$ is the angle in radians.

Q: What are the Addition Formulas for Sine and Cosine?

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$.

Q: What are the three forms of the Double Angle Formula for $\cos 2A$?

$\cos 2A = \cos^2 A - \sin^2 A$, $\cos 2A = 2 \cos^2 A - 1$, and $\cos 2A = 1 - 2 \sin^2 A$.

Q: What is the Double Angle Formula for $\tan 2A$?

$\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$

Q: What is the general strategy for solving complex trigonometric equations in P2?

Use identities to ensure all terms have the same angle (e.g., $\theta$) and the same function (e.g., $\cos$), then solve the resulting quadratic equation.

Q: How do you find the value of $R$ when converting to the Harmonic Form $R \cos(\theta \pm \alpha)$?

$R = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the coefficients of the cosine and sine terms.

Q: How do you calculate the phase shift $\alpha$ in the R-formula?

By comparing the expanded addition formula to the original expression, we find that $\tan \alpha = \frac{b}{a}$ (or $\frac{a}{b}$ depending on the target form).

Q: What are the maximum and minimum values of the expression $a \cos \theta + b \sin \theta$?

The maximum value is $R$ and the minimum value is $-R$, where $R = \sqrt{a^2 + b^2}$.

Practice

Pure Mathematics (YPM01) · Pearson Edexcel International A Level · 9 min read · Updated: 6 Sept 2026

Welcome to P2 Trigonometry: Expanding Your Toolkit!

Hello future mathematician! This chapter, "Trigonometry," is where we take the basic ideas from P1 (sine, cosine, tangent graphs, and solving simple equations) and elevate them using powerful new identities. If you found the trigonometric graphs confusing before, don't worry! We break down these new formulas into simple, easy-to-use tools.

Why is this important? These identities allow us to simplify incredibly complex trigonometric expressions, solve equations that look impossible, and model real-world phenomena like sound waves, light, and tidal movements with greater accuracy. Mastering P2 trigonometry is essential for success in P3.

Section 1: Radian Measure Refresher

In P2, we almost always work with Radians rather than degrees, especially when dealing with calculus (which is coming soon!). Remember, $\pi$ radians is the same as $180^\circ$.

Quick Conversion Formulas

Degrees to Radians: Multiply by $\frac{\pi}{180}$
Radians to Degrees: Multiply by $\frac{180}{\pi}$

Key Concept: Arc Length and Sector Area

These formulas are crucial and only work if the angle $\theta$ is measured in radians.

Arc Length (s): The distance around the curve of the circle sector.
$$s = r\theta$$
Area of Sector (A): The area inside the wedge.
$$A = \frac{1}{2}r^2\theta$$

Quick Review: If a circle has a radius of 5 cm and the angle is $\frac{\pi}{3}$ radians:
Arc Length $s = 5 \times \frac{\pi}{3} = \frac{5\pi}{3}$ cm.
Area $A = \frac{1}{2} \times 5^2 \times \frac{\pi}{3} = \frac{25\pi}{6}$ cm².

Section 2: The Addition (Compound Angle) Formulas

The addition formulas tell us how to find the sine, cosine, or tangent of the sum or difference of two angles, $A$ and $B$. These are the foundation for the rest of P2 trigonometry.

1. Sine Formulas

The pattern for sine involves mixing sine and cosine functions. The sign (plus or minus) stays the same.

$$ \sin(A + B) \equiv \sin A \cos B + \cos A \sin B \ \sin(A - B) \equiv \sin A \cos B - \cos A \sin B $$

Memory Aid for Sine: S. C. C. S. (Sine, Cosine, Cosine, Sine). The sign matches the bracket sign.

2. Cosine Formulas

The pattern for cosine involves grouping the functions (cos-cos, sin-sin). Crucially, the sign flips!

$$ \cos(A + B) \equiv \cos A \cos B - \sin A \sin B \ \cos(A - B) \equiv \cos A \cos B + \sin A \sin B $$

Memory Aid for Cosine: C. C. S. S. (Cosine, Cosine, Sine, Sine). The sign is the opposite of the bracket sign.

3. Tangent Formulas

Tangent is slightly different. Look at the patterns:

$$ \tan(A + B) \equiv \frac{\tan A + \tan B}{1 - \tan A \tan B} \ \tan(A - B) \equiv \frac{\tan A - \tan B}{1 + \tan A \tan B} $$

Memory Aid for Tangent: The sign on the top matches the bracket. The sign on the bottom is the opposite.

⚠ Key Takeaway: Using the Addition Formulas

We use these formulas primarily to:

Find exact values for angles that aren't common, e.g., finding $\sin(15^\circ)$ by calculating $\sin(45^\circ - 30^\circ)$.
Reverse the process: If you see $\cos 3x \cos x - \sin 3x \sin x$, you immediately know this simplifies to $\cos(3x + x) = \cos 4x$.

Section 3: The Double Angle Formulas

The Double Angle Formulas are special cases of the Addition Formulas where $A = B$. These are used constantly in P2 and P3 for solving equations and integration.

1. Sine Double Angle

We start with $\sin(A + B)$ and set $B = A$:

$$ \sin(A + A) \equiv \sin A \cos A + \cos A \sin A $$

This gives the only form for $\sin 2A$:

$$ \sin 2A \equiv 2 \sin A \cos A $$

2. Cosine Double Angle

Start with $\cos(A + B)$ and set $B = A$:

$$ \cos(A + A) \equiv \cos A \cos A - \sin A \sin A $$

This gives the first form for $\cos 2A$:

$$ \cos 2A \equiv \cos^2 A - \sin^2 A $$

Wait, there's more! We know the fundamental P1 identity: $\sin^2 A + \cos^2 A \equiv 1$. We can substitute this into the formula above to get two alternative, very useful forms:

Replace $\cos^2 A$ with $(1 - \sin^2 A)$: $$\cos 2A \equiv (1 - \sin^2 A) - \sin^2 A$$ $$\cos 2A \equiv 1 - 2 \sin^2 A$$
Replace $\sin^2 A$ with $(1 - \cos^2 A)$: $$\cos 2A \equiv \cos^2 A - (1 - \cos^2 A)$$ $$\cos 2A \equiv 2 \cos^2 A - 1$$

Why three forms for $\cos 2A$? When solving trigonometric equations, you must choose the form that results in an equation containing only one type of trig function (either sine or cosine). This is crucial!

3. Tangent Double Angle

Start with $\tan(A + B)$ and set $B = A$:

$$ \tan 2A \equiv \frac{2 \tan A}{1 - \tan^2 A} $$

☢ Common Mistake Alert!

Do NOT confuse $\sin 2A$ with $2 \sin A$. They are completely different.
Example: If $A=30^\circ$, $\sin 2A = \sin 60^\circ = \frac{\sqrt{3}}{2}$. But $2 \sin A = 2 \times \sin 30^\circ = 2 \times \frac{1}{2} = 1$.

Section 4: Solving Trigonometric Equations (P2 Style)

P2 equations often require you to use an identity before you can solve them using standard P1 methods. The goal is always to reduce the equation to a quadratic in a single trigonometric function (e.g., $ax^2 + bx + c = 0$, where $x$ might be $\cos \theta$).

Step-by-Step Example Process

Solve $2 \sin^2 \theta = 3 \cos \theta + 3$ for $0^\circ \le \theta < 360^\circ$.

Step 1: Identify the problem. We have both $\sin^2 \theta$ and $\cos \theta$. We must convert one into the other using the P1 identity $\sin^2 \theta + \cos^2 \theta = 1$.

Step 2: Substitute. Replace $\sin^2 \theta$ with $(1 - \cos^2 \theta)$.

$$ 2(1 - \cos^2 \theta) = 3 \cos \theta + 3 $$

Step 3: Rearrange into a quadratic. Expand and move all terms to one side (keeping the squared term positive is often helpful).

$$ 2 - 2 \cos^2 \theta = 3 \cos \theta + 3 \ 0 = 2 \cos^2 \theta + 3 \cos \theta + 1 $$

Step 4: Solve the quadratic. Let $x = \cos \theta$. Solve $2x^2 + 3x + 1 = 0$.

$$ (2x + 1)(x + 1) = 0 $$

So, $\cos \theta = -1$ or $\cos \theta = -\frac{1}{2}$.

Step 5: Find the angles. Use the inverse function and the CAST diagram or graph knowledge to find all solutions in the given range.

Case 1: $\cos \theta = -1$. $\theta = 180^\circ$
Case 2: $\cos \theta = -\frac{1}{2}$. (Reference angle is $60^\circ$). Cosine is negative in the 2nd and 3rd quadrants.
- 2nd Q: $180^\circ - 60^\circ = 120^\circ$
- 3rd Q: $180^\circ + 60^\circ = 240^\circ$

Solutions: $120^\circ, 180^\circ, 240^\circ$.

✅ P2 Identity Strategy Checklist

When solving an equation:

If you have different angles ($2\theta$ and $\theta$), use a Double Angle formula to make them the same ($\theta$).
If you have different functions ($\sin^2$ and $\cos$), use the Pythagorean Identity ($\sin^2 x + \cos^2 x = 1$) to make them the same function.

Section 5: The R-Formula (The Harmonic Form)

This is arguably the most important new topic in P2 Trigonometry. The R-formula takes an expression containing both sine and cosine terms (like $a \cos \theta + b \sin \theta$) and converts it into a single, simpler wave function (like $R \cos(\theta - \alpha)$).

Purpose: To easily find the maximum and minimum values of the expression, and to solve equations involving combined sine and cosine functions.

We are aiming to prove that:

$$\na \cos \theta + b \sin \theta \equiv R \cos(\theta - \alpha) $$

...or one of its three variations ($R \sin(\theta + \alpha)$, $R \cos(\theta + \alpha)$, $R \sin(\theta - \alpha)$).

Analogy: Imagine two different sound waves combining (sine and cosine). The R-formula shows you that the result is just a single, new wave (R-wave) that is slightly louder (R) and slightly shifted ($\alpha$).

Step-by-Step: Finding R and $\alpha$

Let's find $R$ and $\alpha$ for the expression $3 \cos \theta + 4 \sin \theta$, converting it to the form $R \cos(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$.

Step 1: Expand the R-Form.

Use the Addition Formula for $\cos(\theta - \alpha)$:

$$\nR \cos(\theta - \alpha) \equiv R (\cos \theta \cos \alpha + \sin \theta \sin \alpha) \\nR \cos(\theta - \alpha) \equiv (R \cos \alpha) \cos \theta + (R \sin \alpha) \sin \theta $$

Step 2: Compare Coefficients.

Compare this expanded form with the original expression: $3 \cos \theta + 4 \sin \theta$.

Coefficient of $\cos \theta$: $R \cos \alpha = 3$ (Equation 1)
Coefficient of $\sin \theta$: $R \sin \alpha = 4$ (Equation 2)

Step 3: Find R (The Magnitude).

Square both equations and add them together:

$$ (R \cos \alpha)^2 + (R \sin \alpha)^2 = 3^2 + 4^2 \\nR^2 (\cos^2 \alpha + \sin^2 \alpha) = 9 + 16 \\nR^2 (1) = 25 \\nR = 5 $$

Formula for R: $R = \sqrt{a^2 + b^2}$ (It's simply Pythagoras!).

Step 4: Find $\alpha$ (The Phase Shift).

Divide Equation 2 by Equation 1:

$$ \frac{R \sin \alpha}{R \cos \alpha} = \frac{4}{3} \ \tan \alpha = \frac{4}{3} \ \alpha = \arctan\left(\frac{4}{3}\right) \approx 53.1^\circ \text{ (to 1 d.p.)} $$

Conclusion: $3 \cos \theta + 4 \sin \theta \equiv 5 \cos(\theta - 53.1^\circ)$.

Applications of the R-Formula

Once you have the R-form, you can easily determine:

1. Maximum and Minimum Values

Since $-1 \le \cos(\theta - \alpha) \le 1$, the range of the whole expression $R \cos(\theta - \alpha)$ is:

Maximum Value: $R$
Minimum Value: $-R$

In our example, the maximum value is 5, and the minimum value is -5.

2. Solving Equations

To solve $3 \cos \theta + 4 \sin \theta = 2$, you would rewrite it as:

$$ 5 \cos(\theta - 53.1^\circ) = 2 \ \cos(\theta - 53.1^\circ) = 0.4 $$

This is now a standard P1-style equation, making the solving process much simpler.

Did you know? (The P2 Wave)

The R-Formula is sometimes called the Harmonic Form. It proves that any combination of sine and cosine waves of the same frequency is equivalent to a single wave (the resulting wave), demonstrating a fundamental principle in physics and signal processing!

Chapter Summary and Key Takeaways

This P2 chapter provides powerful tools for manipulating and solving trigonometric problems. Focus on these main goals:

Radians: Ensure you use $s=r\theta$ and $A=\frac{1}{2}r^2\theta$ correctly, with $\theta$ in radians.
Identities: Memorise the Addition Formulas (S.C.C.S. and C.C.S.S.).
Double Angle: Know the three forms of $\cos 2A$. Select the correct one to create an equation in a single function (e.g., only $\sin \theta$).
R-Formula: Master the four steps: Expand, Compare, Find $R$ (Pythagoras), Find $\alpha$ ($\tan \alpha$).

Don't worry if the identities look intimidating at first. Write them out multiple times, practice recognizing the reverse forms, and you will soon feel confident using them!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

How do you convert between degrees and radians?

To convert degrees to radians, multiply by $\frac{\pi}{180}$; to convert radians to degrees, multiply by $\frac{180}{\pi}$.

What is the formula for the arc length of a circle sector?

The arc length $s$ is given by $s = r\theta$, where $r$ is the radius and $\theta$ is the angle in radians.

What is the formula for the area of a circle sector?

The area $A$ is given by $A = \frac{1}{2}r^2\theta$, where $r$ is the radius and $\theta$ is the angle in radians.

What are the Addition Formulas for Sine and Cosine?

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ and $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$.

What is the Double Angle Formula for $\sin 2A$?

$\sin 2A = 2 \sin A \cos A$

What are the three forms of the Double Angle Formula for $\cos 2A$?

$\cos 2A = \cos^2 A - \sin^2 A$, $\cos 2A = 2 \cos^2 A - 1$, and $\cos 2A = 1 - 2 \sin^2 A$.

What is the Double Angle Formula for $\tan 2A$?

$\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$

What is the general strategy for solving complex trigonometric equations in P2?

Use identities to ensure all terms have the same angle (e.g., $\theta$) and the same function (e.g., $\cos$), then solve the resulting quadratic equation.

How do you find the value of $R$ when converting to the Harmonic Form $R \cos(\theta \pm \alpha)$?

$R = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the coefficients of the cosine and sine terms.

How do you calculate the phase shift $\alpha$ in the R-formula?

By comparing the expanded addition formula to the original expression, we find that $\tan \alpha = \frac{b}{a}$ (or $\frac{a}{b}$ depending on the target form).

What are the maximum and minimum values of the expression $a \cos \theta + b \sin \theta$?

The maximum value is $R$ and the minimum value is $-R$, where $R = \sqrt{a^2 + b^2}$.

Ready to test yourself?

7 practice questions on this topic, with worked solutions. Multiple choice is marked as you go. No account needed.

Practise This Topic

More Pure Mathematics (YPM01) chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practise This Topic Now