O-Level Add Maths: Trigonometric Identities and Equations

Q: What are the 3 fundamental trig identities?

1) sin^2(x) + cos^2(x) = 1 (the Pythagorean Identity). 2) tan(x) = sin(x)/cos(x). 3) 1 + tan^2(x) = sec^2(x). The first two are tested in almost every single CAIE paper.

Q: What is the CAST diagram?

CAST tells you which trig ratios are positive in each quadrant. Going anticlockwise from the 4th quadrant: C (Cos positive, Q4), A (All positive, Q1), S (Sin positive, Q2), T (Tan positive, Q3). This is essential for finding all solutions in a given range.

Q: How do I solve sin(x) = 0.5 for 0 to 360 degrees?

Step 1: Find the principal value: x = sin^(-1)(0.5) = 30 degrees. Step 2: Sin is positive, so use CAST: sin is positive in Q1 and Q2. Step 3: Q1 solution = 30 degrees. Q2 solution = 180 - 30 = 150 degrees. Answer: x = 30 or 150 degrees.

Q: How do I prove a trig identity?

Work on ONE side only (usually the more complex side). Use substitutions like tan = sin/cos and sin^2+cos^2=1 to simplify it until it matches the other side. Never cross the equals sign to work on both sides simultaneously.

A unit circle diagram with quadrant labels and trigonometric values.

What is the single most important trig identity?

sin^2(x) + cos^2(x) = 1. This Pythagorean Identity is used in virtually every trig proof and equation. If you need sin^2(x), rearrange to get sin^2(x) = 1 - cos^2(x). If you need cos^2(x), rearrange to cos^2(x) = 1 - sin^2(x). Memorize this the way you memorize your own name — it appears in every single CAIE exam paper.

1. The 3 Core Identities
2. The CAST Diagram (Quadrant Rule)
3. Solving Trig Equations
4. Proving Trigonometric Identities

Trigonometric Identities are the most algebra-heavy topic in Add Maths. Raw memorization gets you nowhere. You need to understand why the identities work and develop the instinct for which substitution to make. This guide from our Ultimate Add Maths Guide builds that instinct.

1. The 3 Core Identities

Identity 1: The Pythagorean Identity

sin²(x) + cos²(x) = 1

Identity 2: The Tangent Ratio

tan(x) = sin(x) / cos(x)

Identity 3: The Secant Identity

1 + tan²(x) = sec²(x)

💡 Tutor's Tip

Derivation Trick: Identity 3 is just Identity 1 divided by cos²(x). Divide every term: sin²/cos² + cos²/cos² = 1/cos² gives you tan²(x) + 1 = sec²(x). Understanding the derivation means you never need to memorize it separately.

2. The CAST Diagram (Quadrant Rule)

When solving trig equations, your calculator only gives you ONE answer (the principal value). But there are usually TWO solutions in the range 0° to 360°. The CAST diagram tells you where to find the second one.

Q2: S (only Sin positive) | Q1: A (All positive)

Q3: T (only Tan positive) | Q4: C (only Cos positive)

Finding the Second Solution

Sin positive: Q1 = α, Q2 = 180° - α

Cos positive: Q1 = α, Q4 = 360° - α

Tan positive: Q1 = α, Q3 = 180° + α

3. Solving Trig Equations

Worked: Solve 2sin²(x) - sin(x) - 1 = 0 for 0° ≤ x ≤ 360°

Step 1: Let u = sin(x). Equation becomes 2u² - u - 1 = 0.

Step 2: Factorize: (2u + 1)(u - 1) = 0. So u = -1/2 or u = 1.

Step 3: sin(x) = 1 gives x = 90°.

Step 4: sin(x) = -0.5. Principal value = -30°. Sin is negative in Q3 and Q4. Q3: 180+30 = 210°. Q4: 360-30 = 330°. x = 90°, 210°, 330°

💡 Tutor's Tip

The Substitution Technique: When you see sin² + cos² in an equation, replace one with the other using Identity 1 to reduce the equation to a single trig function. Then it becomes a normal quadratic that you can factorize.

4. Proving Trigonometric Identities

In a "Prove that..." question, you must start from ONE side of the equation (usually the more complex side) and manipulate it algebraically until it looks exactly like the other side. You must NEVER work on both sides simultaneously.

The 3 Go-To Substitutions

1. Replace tan(x) with sin(x)/cos(x) whenever you see tan.

2. Replace sin²(x) with 1 - cos²(x) (or vice versa) to eliminate a variable.

3. Create a common denominator if you see fractions, then simplify.

📋 From the Desk of Joshua Ramirez

The Proof Layout:Start by writing "LHS =" (Left Hand Side), then show every single algebraic manipulation step underneath. End with "= RHS". Never write "LHS = RHS" at the start — that is what you are trying to prove, and assuming it is circular logic that earns zero marks.

Frequently Asked Questions

What are the 3 fundamental trig identities?▼

sin^2+cos^2=1, tan=sin/cos, and 1+tan^2=sec^2. The first two handle 95% of CAIE questions.

What is the CAST diagram?▼

A quadrant map showing which trig ratios are positive: All in Q1, Sin in Q2, Tan in Q3, Cos in Q4.

How do I solve sin(x) = 0.5 for 0 to 360 degrees?▼

Principal value = 30 degrees. Sin positive in Q1 and Q2. Solutions: 30 degrees and 180-30 = 150 degrees.

How do I prove a trig identity?▼

Work on one side only. Use substitutions (tan=sin/cos, sin^2=1-cos^2) to algebraically transform it into the other side.

Stop Guessing, Start Scoring

500+ CAIE practice questions with worked solutions and AI-powered mock exams.

Access the Study Portal →Prepare with the Oracle Engine

The Angle Arsenal: Mastering Trigonometric Identities