IGCSE Trigonometry Equation Solver

Solve sin(ax) = k, cos(ax) = k, tan(ax) = k using CAST diagram / quadrants. Enter the coefficient, value and domain to get all solutions with clear working.

Current Equation
sin(1x) = 0.5
Domain Limit: -3600° to 3600°
How to enter radians with π in domain section:
You can type pi, 2pi, pi/2, 3*pi, -pi/3, etc.
Pure numbers (e.g. 3.14) are also accepted. This works when the unit is set to Radians.
Exam Tip: Always find the principal (acute) angle first, then use the CAST diagram to locate all solutions in the correct quadrants. Finally filter the answers that lie inside the given domain. Remember: sin is positive in 1st & 2nd, cos in 1st & 4th, tan in 1st & 3rd.

How to use this solver

  1. Choose the trigonometric function (sin / cos / tan).
  2. Enter the coefficient a (for example if the equation is sin(2x) = 0.5, then a = 2).
  3. Enter the value of k (the right-hand side).
  4. Set the domain (e.g. 0° to 360° or –180° to 180°).
  5. Click the button to get all solutions with CAST working.

Understanding Quadrants (CAST Diagram)

In IGCSE Mathematics, we use the CAST diagram (also called the ASTC diagram) to determine where sine, cosine and tangent are positive or negative. This is essential when solving trigonometric equations.

CAST Rule

  • C – Cosine positive in 4th quadrant
  • A – All positive in 1st quadrant
  • S – Sine positive in 2nd quadrant
  • T – Tangent positive in 3rd quadrant

Quick Reference

  • sin positive: 1st & 2nd
  • cos positive: 1st & 4th
  • tan positive: 1st & 3rd

How to Solve Using Quadrants

  1. Find the principal (acute) angle α using inverse trig function.
  2. Use the CAST diagram to find all angles in the range 0°–360° (or 0–2π) that satisfy the equation.
  3. Write the general solution (add 360°n or 2πn).
  4. If the equation is of the form sin(ax) = k, divide all solutions by a.
  5. Select only the solutions that lie within the given domain.

Example 1

Solve sin(x) = 0.5   for   0° ≤ x ≤ 360°

Solution:
Principal angle α = sin⁻¹(0.5) = 30°
Sine is positive in 1st and 2nd quadrants → x = 30° and x = 180° – 30° = 150°
Answer: x = 30°, 150°

Example 2

Solve cos(2x) = –√3/2   for   0° ≤ x ≤ 180°

Solution:
Principal angle α = cos⁻¹(√3/2) = 30°
Cosine is negative in 2nd and 3rd quadrants → 2x = 180° – 30° = 150° and 2x = 180° + 30° = 210°
Also consider the next cycle if needed: 2x = 360° + 150° = 510°, etc.
Divide by 2: x = 75°, 105°
(Check which values lie in 0° ≤ x ≤ 180°)
Answer: x = 75°, 105°

Example 3 (Radians)

Solve tan(x) = 1   for   0 ≤ x ≤ 2π

Solution:
Principal angle α = tan⁻¹(1) = π/4
Tangent is positive in 1st and 3rd quadrants → x = π/4 and x = π + π/4 = 5π/4
Answer: x = π/4, 5π/4

Advanced Concept: Periodicity & Composite Angles sin(ax)

When solving trigonometric equations where the argument is scaled by a coefficient a (such as sin(ax), cos(ax), or tan(ax)), the fundamental period of the function changes. This is one of the most common places where IGCSE students miss solutions.

How the Period Changes

The standard period for sin(x) and cos(x) is 360° (or 2π). For sin(ax) and cos(ax), the new period is:
New Period = 360° / a (or 2π / a)
For tan(ax), the period is 180° / a (or π / a).

The Golden Rule for Scaling Domains

Always expand your working domain first! If your given domain is 0° ≤ x ≤ 360° and you are solving for 2x, you must find solutions for ax across an extended range:
0° ≤ ax ≤ 360° × a
Then, divide all valid angles by a at the very end.

Strategy for Solving sin(ax) = k

  1. Let u = ax: Temporarily substitute the inner expression so the equation looks like a standard form (sin(u) = k).
  2. Adjust the Domain: Multiply the lower and upper bounds of your x-domain by a to find the new range for u.
  3. Find all values of u: Use the CAST diagram and principal angle within your extended u-range.
  4. Solve for x: Divide every valid value of u by a.
  5. Filter: Retain only the values of x that lie strictly inside the original requested domain.

⚠️ Common Exam Pitfalls & How to Avoid Them

Examiners frequently design questions to catch students who rely on shortcuts rather than understanding the underlying symmetry of trigonometric curves. Keep these warnings in mind:

1. Forgetting to check additional cycles when a > 1

If a = 2 and the domain is up to 360°, your working domain for 2x goes up to 720°. Stopping after the first full rotation means you will miss half of the correct answers!

2. Mixing up Degree and Radian modes

If the question specifies a domain in radians (e.g., 0 ≤ x ≤ 2π), ensure your solutions are expressed in exact radian multiples (like π/3, 3π/4) rather than decimals unless explicitly requested.

3. Ignoring the Range of Sine and Cosine

Remember that sin(x) and cos(x) can never exceed 1 or fall below -1. If an equation results in sin(x) = 1.4, state immediately that "No solution exists" because 1.4 is outside the valid range [-1, 1].

Practice Problem Bank (Try These Yourself)

Test your understanding by attempting these standard IGCSE-style problems, then use the solver above to verify your answers and check your working steps.

Problem 1 (Basic Sine):

Solve sin(x) = -0.4226 for 0° ≤ x ≤ 360°.

Click to reveal hint & answer

Hint: Since sine is negative, look in the 3rd and 4th quadrants.
Answer: x = 205.0°, 335.0°

Problem 2 (Scaled Coefficient):

Solve cos(3x) = 0.5 for 0° ≤ x ≤ 180°.

Click to reveal hint & answer

Hint: Extend your range for 3x up to 180° × 3 = 540°.
Answer: x = 20.0°, 100.0°, 140.0°

Problem 3 (Radian Domain):

Solve tan(x) = -√3 for 0 ≤ x ≤ 2π.

Click to reveal hint & answer

Hint: Principal angle is π/3. Tangent is negative in the 2nd and 4th quadrants.
Answer: x = 2π/3, 5π/3

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