Trigonometric Equations Explained — How to Solve Them (JAMB/WAEC Mathematics)

A trigonometric equation is an equation involving a trigonometric ratio (sine, cosine, or tangent) that you must solve for the unknown angle, usually within a given range. This is the next step in our Trigonometry series, building on the ratios, identities, and formulas covered in earlier lessons.

This topic is a regular feature in JAMB and WAEC Mathematics, usually tested by asking students to find all solutions within a specified interval, such as 0° to 360°. This lesson explains the method step by step with worked examples.

Quick takeaways

• A trigonometric equation asks you to find the angle (or angles) that satisfy a given trigonometric ratio, usually within a specified range like 0° to 360°.
• Trigonometric ratios repeat in a pattern across four quadrants, which means most trigonometric equations have more than one solution within a full circle.
• The CAST rule (or ASTC) helps determine which quadrants give a positive value for a specific ratio: All ratios positive in quadrant 1, Sine positive in quadrant 2, Tangent positive in quadrant 3, Cosine positive in quadrant 4.
• To find all solutions, first find the basic (reference) angle using inverse trigonometric functions, then use the CAST rule to find the corresponding angles in the correct quadrants.
• Some trigonometric equations are quadratic in form and require factoring or substitution before applying the standard solving method.

What is a trigonometric equation?

A trigonometric equation is an equation containing a trigonometric ratio (such as sinθ, cosθ, or tanθ) set equal to a specific value, which you must solve to find the angle or angles θ that make the equation true. Unlike a simple algebraic equation, trigonometric equations often have multiple solutions within a given range, since trigonometric ratios repeat their values in a predictable pattern as the angle increases through a full circle.

What is the CAST rule, and why is it important?

The CAST rule (also written as ASTC) describes which trigonometric ratios are positive in each of the four quadrants of a circle:

• Quadrant 1 (0°–90°): All ratios are positive
• Quadrant 2 (90°–180°): Sine is positive (cosine and tangent are negative)
• Quadrant 3 (180°–270°): Tangent is positive (sine and cosine are negative)
• Quadrant 4 (270°–360°): Cosine is positive (sine and tangent are negative)

This rule is essential for trigonometric equations because it tells you which quadrants to look in for additional solutions, beyond the first basic angle you calculate.

How do you solve a basic trigonometric equation?

To solve an equation like sinθ = 0.5 for 0° ≤ θ ≤ 360°:

1. Find the basic (reference) angle using the inverse function: θ = sin⁻¹(0.5) = 30°
2. Identify which quadrants give a positive value for sine, using the CAST rule — in this case, quadrants 1 and 2
3. Find the angle in each relevant quadrant:
   • Quadrant 1: θ = 30° (the basic angle itself)
   • Quadrant 2: θ = 180° − 30° = 150°
4. State both solutions: θ = 30° or θ = 150°

The specific formula for finding the angle in each quadrant differs slightly depending on which quadrant you’re working in, so it’s important to learn the correct adjustment for each one (e.g., 180° − basic angle for quadrant 2, 180° + basic angle for quadrant 3, 360° − basic angle for quadrant 4).

How do you solve a quadratic trigonometric equation?

Some trigonometric equations are quadratic in form, such as 2sin²θ − sinθ − 1 = 0. These are solved by treating the trigonometric ratio as if it were a single variable:

1. Substitute a variable (e.g., let x = sinθ) to turn the equation into a standard quadratic: 2x² − x − 1 = 0
2. Solve the quadratic equation for x using factoring or the quadratic formula
3. Substitute back to solve for θ using each valid value of x (remembering that sinθ, cosθ must stay between −1 and 1)
4. Apply the CAST rule as usual to find all solutions within the given range

Common mistakes students make with trigonometric equations

• Finding only the basic angle and forgetting to check for additional solutions in other quadrants
• Misapplying the CAST rule, or forgetting which ratio is positive in which quadrant
• Using the wrong adjustment formula for a specific quadrant (e.g., using 180° − angle when 360° − angle was needed)
• In quadratic trigonometric equations, forgetting to check that a solution for sinθ or cosθ falls within the valid range of −1 to 1

Frequently asked questions

Q: What is the CAST rule in trigonometry? The CAST rule describes which trigonometric ratios are positive in each of the four quadrants: All ratios are positive in quadrant 1, Sine in quadrant 2, Tangent in quadrant 3, and Cosine in quadrant 4.

Q: Why do trigonometric equations often have more than one solution? Because trigonometric ratios repeat their values in a predictable pattern across the four quadrants of a full circle, so more than one angle within a given range can produce the same ratio value.

Q: How do you solve a trigonometric equation that is quadratic in form? Substitute the trigonometric ratio with a single variable to form a standard quadratic equation, solve for that variable using factoring or the quadratic formula, then substitute back and apply the CAST rule to find all valid angles.

Q: What should you check before accepting a solution to a quadratic trigonometric equation? Check that any solution for sinθ or cosθ falls within the valid range of −1 to 1, since values outside this range are not possible for these ratios.

New to trigonometry? Start with Trigonometry Basics →, then identities and complementary angles →, and sum and difference of angles →

This is one topic from our full JAMB & WAEC Maths course — see everything covered →