Trigonometry

Introduction

Assalamu alaikum, students. I am Ustad Bilal Ahmed, and having marked many Cambridge A Level scripts over the years here in Lahore, I can tell you with certainty that Trigonometry is one of the most foundational and high-yield topics you will encounter. It forms a significant portion of your Pure Mathematics 1 (P1) paper and is then extended into more complex applications in Pure Mathematics 3 (P3).

Mastery of trigonometry is not just about passing your exam; it is the language of engineering, physics, architecture, and even computer graphics. Whether you dream of studying at NUST, GIKI, or LUMS, the principles you learn here – manipulating waves, analysing forces, and describing rotations – will be invaluable. In this chapter, we will build your skills from the ground up, from radians and basic triangle rules to solving sophisticated trigonometric equations. Pay close attention, practice diligently, and you will find these questions become a reliable source of marks, insha'Allah.

Core Theory

Let's begin with the fundamentals and build our way up to the more advanced P3 concepts.

#### 1. Radian Measure

While you are familiar with degrees, the natural unit for measuring angles in A Level Maths and beyond is the radian. One radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. The key conversion is:

π radians = 180°

This leads to two essential formulae for a sector of a circle with radius *r* and angle *θ* (in radians):

• Arc Length (s): `s = rθ`
• Area of Sector (A): `A = ½r²θ`

*Example:* A circle has radius 5 cm. Find the arc length and area of a sector with an angle of 2π/5 radians.

• Arc Length: `s = 5 * (2π/5) = 2π cm`
• Sector Area: `A = ½ * 5² * (2π/5) = ½ * 25 * (2π/5) = 5π cm²`

Always ensure your calculator is in the correct mode (RAD/DEG)! This is the most common mistake I see.

#### 2. Sine and Cosine Rule

For any non-right-angled triangle ABC:

• Sine Rule: Used when you have a side and its opposite angle.

`a/sin A = b/sin B = c/sin C`

• Cosine Rule: Used when you have two sides and the included angle (SAS) or all three sides (SSS).

`a² = b² + c² - 2bc cos A` (to find a side)

`cos A = (b² + c² - a²) / 2bc` (to find an angle)

• Area of a Triangle: `Area = ½ab sin C`

#### 3. Core Identities & Exact Values

You must memorise the exact values for special angles. No excuses!

| Angle (θ) | 30° (π/6) | 45° (π/4) | 60° (π/3) |

|-----------|-----------|-----------|-----------|

| sin θ | 1/2 | 1/√2 | √3/2 |

| cos θ | √3/2 | 1/√2 | 1/2 |

| tan θ | 1/√3 | 1 | √3 |

The fundamental identities are your main tools for simplification and solving equations:

• `tanθ ≡ sinθ/cosθ`
• `sin²θ + cos²θ ≡ 1`

From this, we derive the P3 identities by dividing by `cos²θ` and `sin²θ` respectively:

• `1 + tan²θ ≡ sec²θ`
• `cot²θ + 1 ≡ cosec²θ`

#### 4. Compound and Double Angle Formulae

These are given in the formula book (MF19), but you must know how to use them fluently.

• Compound Angles:
• `sin(A ± B) = sinAcosB ± cosAsinB`
• `cos(A ± B) = cosAcosB ∓ sinAsinB` (note the sign change!)
• `tan(A ± B) = (tanA ± tanB) / (1 ∓ tanAtanB)`

• Double Angles: (Derived by setting B=A in the above)
• `sin(2A) = 2sinAcosA`
• `cos(2A) = cos²A - sin²A`
• This has two other crucial forms, derived using `sin²A + cos²A = 1`:
• `cos(2A) = 2cos²A - 1`
• `cos(2A) = 1 - 2sin²A`
• `tan(2A) = 2tanA / (1 - tan²A)`

Knowing which form of `cos(2A)` to use is key. For example, if your equation also contains a `sinA` term, use the `1 - 2sin²A` form to create a quadratic in sine.

#### 5. The R-Formula: `a sin x ± b cos x`

A common P3 question asks you to express `a sin x ± b cos x` in the form `R sin(x ± α)` or `R cos(x ± α)`. Let's take `a sin x + b cos x = R sin(x + α)`.

By expanding the right side using the compound angle formula, we get `R(sinxcosα + cosxsinα)`.

Comparing coefficients:

• `a = Rcosα`
• `b = Rsinα`

To find R and α:

• Square and add: `a² + b² = R²(cos²α + sin²α) = R²`, so `R = √(a² + b²)`.
• Divide: `(Rsinα)/(Rcosα) = b/a`, so `tanα = b/a`.

This form is useful for finding the maximum/minimum value of the expression (which is simply ±R) and for solving equations.

#### 6. Solving Equations

This is a process that brings everything together.

1. Simplify: Use identities to simplify the equation into one involving a single trigonometric function, e.g., only `sin x`. This often involves creating a quadratic.
2. Solve: Solve for the trig function, e.g., find the value of `sin x`.
3. Principal Value: Find the first angle from your calculator, `α = sin⁻¹(value)`.
4. All Solutions: Use a CAST diagram or graph sketch to find all other solutions within the required interval (e.g., 0° to 360° or 0 to 2π).
5. Check: Ensure your answers are in the correct format (degrees/radians) and within the specified range.

Key Definitions

• Radian: The angle at the centre of a circle where the arc length is equal to the radius. π radians = 180°.
• Arc Length: `s = rθ` (θ in radians).
• Sector Area: `A = ½r²θ` (θ in radians).
• Sine Rule: `a/sin A = b/sin B = c/sin C`.
• Cosine Rule: `a² = b² + c² - 2bc cos A`.
• Pythagorean Identity: `sin²θ + cos²θ ≡ 1`.
• Secant (sec), Cosecant (cosec), Cotangent (cot): `secθ = 1/cosθ`, `cosecθ = 1/sinθ`, `cotθ = 1/tanθ`.
• Double Angle Formulae: `sin(2A) = 2sinAcosA`, `cos(2A) = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A`.
• R-Formula: An expression `a sin x + b cos x` can be written as `R sin(x + α)` where `R = √(a²+b²)` and `tan α = b/a`.

Worked Examples (Pakistani Context)

#### Example 1: Solving a P3-style Equation

Question:

a) Express `cos(2x) + 3sin(x)` in terms of `sin(x)` only.

b) Hence, solve the equation `cos(2x) + 3sin(x) - 2 = 0` for `0° ≤ x ≤ 360°`.

Solution:

a) We need to replace `cos(2x)`. Since the other term is `sin(x)`, we should choose the form of `cos(2x)` that involves sine.

Use `cos(2x) ≡ 1 - 2sin²(x)`.

So, `cos(2x) + 3sin(x) = (1 - 2sin²(x)) + 3sin(x)`.

This is `-2sin²(x) + 3sin(x) + 1`. [B1 mark for correct substitution]

b) Using the result from part (a), the equation becomes:

`-2sin²(x) + 3sin(x) + 1 - 2 = 0`

`-2sin²(x) + 3sin(x) - 1 = 0`

Multiply by -1 to make it easier to factorise:

`2sin²(x) - 3sin(x) + 1 = 0` [M1 for forming a quadratic in sin(x)]

Let `y = sin(x)`. The equation is `2y² - 3y + 1 = 0`.

Factoring gives `(2y - 1)(y - 1) = 0`.

So, `y = 1/2` or `y = 1`.

This means `sin(x) = 1/2` or `sin(x) = 1`. [A1 for correct solutions for sin(x)]

Case 1: `sin(x) = 1/2`

Principal value: `x = sin⁻¹(0.5) = 30°`.

Sine is positive in the 1st and 2nd quadrants.

Second solution is `180° - 30° = 150°`.

Case 2: `sin(x) = 1`

From the graph of sin(x), we know this occurs at `x = 90°`.

So, the final solutions in the range `0° ≤ x ≤ 360°` are 30°, 90°, 150°. [A1 for all correct solutions]

#### Example 2: Lahore Orange Line Metro Track

Question:

A curved section of the Lahore Orange Line track between two points, A and B, forms an arc of a circle with its centre, O, at a major interchange. Surveyors place a coordinate system with O at the origin (0,0). Point A is at (300, 400) and point B is at (400, 300), with distances in metres.

a) Calculate the radius of the curve.

b) Find the angle AOB in radians.

c) Calculate the length of the track between A and B.

Solution:

a) The radius of the circle is the distance from the centre O(0,0) to any point on the circumference, like A.

Using the distance formula:

`r = OA = √( (300-0)² + (400-0)² )`

`r = √(90000 + 160000) = √250000 = 500` m.

The radius of the curve is 500 m. [M1 A1]

b) To find the angle AOB, we can use the Cosine Rule on triangle AOB. We have the lengths of OA = 500m and OB = 500m. We need the length of the chord AB.

Distance AB = `√( (400-300)² + (300-400)² )`

AB = `√( 100² + (-100)² ) = √(10000 + 10000) = √20000` m.

Now, using the Cosine Rule to find angle θ = AOB:

`AB² = OA² + OB² - 2(OA)(OB)cos(θ)`

`20000 = 500² + 500² - 2(500)(500)cos(θ)`

`20000 = 250000 + 250000 - 500000cos(θ)`

`20000 = 500000 - 500000cos(θ)`

`500000cos(θ) = 480000`

`cos(θ) = 480000 / 500000 = 0.96` [M1 for correct application of Cosine Rule]

`θ = cos⁻¹(0.96)`. Make sure calculator is in RADIAN mode.

`θ ≈ 0.2838` radians (to 4 s.f.). [A1]

c) The length of the track is the arc length `s = rθ`.

`s = 500 * 0.2838`

`s ≈ 141.9` m.

The length of the track is 142 m (to 3 s.f.). [M1 A1]

Exam Technique

As an examiner, I notice the same mistakes year after year. Here is how you can avoid them, beta.

1. Degrees vs. Radians: This is the #1 silly mistake. Before you start any question, look at the interval given (e.g., 0 to 2π or 0° to 360°). Set your calculator to the correct mode IMMEDIATELY. For questions involving arc length or sector area, you must use radians.

1. 'Show That' Questions: These are a gift! The answer is given. Your job is to produce a clear, logical argument from the starting expression to the target expression. Do not skip steps. For example, if you use `sin²θ + cos²θ = 1`, write it down. The examiner cannot read your mind.

1. Finding ALL Solutions: When you solve an equation like `sin(x) = 0.5`, your calculator gives you one answer (the principal value). You must use the CAST diagram or a graph sketch to find all other solutions in the required range. Forgetting the second, third, or fourth solution is a very common way to lose marks.

1. Mark Scheme Insights:

• M1 (Method Mark): Awarded for attempting to use a correct method. For example, using the quadratic formula or applying a correct identity. You get this even if your algebra is wrong.
• A1 (Accuracy Mark): Awarded for a correct answer or correct intermediate step. You can only get this if the preceding M mark was earned.
• B1 (Independent Mark): Awarded for a correct statement, formula, or graph, independent of method.

1. Premature Approximation: Do not round your intermediate values. If you find `θ = cos⁻¹(0.4)`, use that exact value in your next calculation. Only round your final answer to the required degree of accuracy (usually 3 significant figures).

By focusing on these details, you show the examiner that you are a careful and competent mathematician. Good presentation and clear working will always be rewarded.