How do I solve sin(2x) = 0.5 for x between 0 and 360 degrees?

First, let u = 2x. The interval for u is 0° ≤ u ≤ 720° (since x goes up to 360°, 2x goes up to 720°). Solve sin u = 0.5: u = 30° and 150° (principal solutions). Then add 360° to each: u = 390° and 510°. So u = 30°, 150°, 390°, 510°. Now divide by 2: x = 15°, 75°, 195°, 255°. Always check that these are within the original interval (0° to 360°).

What is the CAST diagram and how do I use it?

The CAST diagram is a circle divided into four quadrants: C (cos positive) in IV, A (all positive) in I, S (sin positive) in II, T (tan positive) in III. To find all solutions, first find the acute angle (principal value) using inverse trig. Then, for sin and cos, the second solution is 180° minus the acute angle (sin) or 360° minus the acute angle (cos). For tan, add 180° to the acute angle. The diagram helps you remember which quadrants give positive values.

How do I solve a quadratic trig equation like 2 sin²θ - sinθ - 1 = 0?

Treat sinθ as a variable, say x. So 2x² - x - 1 = 0. Factorise: (2x + 1)(x - 1) = 0, so x = -1/2 or x = 1. Thus sinθ = -1/2 or sinθ = 1. Solve each: sinθ = 1 gives θ = 90° (or π/2). sinθ = -1/2: sin is negative in quadrants III and IV. Acute angle = 30°, so solutions are 180°+30°=210° and 360°-30°=330° (or 7π/6 and 11π/6). So final solutions: 90°, 210°, 330° (in degrees).

What if the equation involves tan and a multiple angle, like tan(3θ) = 1 for 0 ≤ θ ≤ 180°?

Let u = 3θ. Interval for u: 0° ≤ u ≤ 540°. Solve tan u = 1: principal solution u = 45°. Since tan repeats every 180°, add 180°: u = 45°, 225°, 405°. Now divide by 3: θ = 15°, 75°, 135°. All are within 0° to 180°. Always check that you haven't exceeded the upper bound; if u went beyond 540°, you'd stop.

How do I handle equations like cos²θ = 1/4?

Take square roots: cosθ = ±1/2. Solve separately. For cosθ = 1/2: solutions are 60° and 300° (or π/3 and 5π/3). For cosθ = -1/2: solutions are 120° and 240° (or 2π/3 and 4π/3). So total four solutions in [0°, 360°). Remember that squaring introduces extraneous solutions? No, because we took square root of both sides, so both signs are valid. Always check the original equation if unsure.

What's the difference between solving in degrees and radians?

The process is identical, but the interval bounds and solutions are in radians. For example, sinθ = 0.5 in [0, 2π): solutions are π/6 and 5π/6. When adjusting intervals, multiply by the same factor. For instance, sin(2θ) = 0.5 in [0, π): let u = 2θ, interval [0, 2π). Solve sin u = 0.5: u = π/6, 5π/6, then add 2π: u = 13π/6, 17π/6. Divide by 2: θ = π/12, 5π/12, 13π/12, 17π/12. But 13π/12 and 17π/12 are > π, so discard? Actually, 13π/12 ≈ 1.083π > π, so only π/12 and 5π/12 are in [0, π). So always check the original interval.

Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle

EDEXCEL

A-Level

This topic focuses on solving trigonometric equations within a specified interval. It covers quadratic equations in terms of sine, cosine, and tangent, as well as equations involving multiples of the unknown angle, requiring students to work in both degrees and radians.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

Topic Overview

This topic covers solving trigonometric equations of the form sin(ax + b) = c, cos(ax + b) = c, tan(ax + b) = c, and quadratic equations in sin, cos, or tan (e.g., 2 sin²θ - sinθ - 1 = 0). You'll learn to find all solutions within a given interval, often using the cast diagram or trigonometric graphs. Mastery here is essential for A-Level Maths and Further Maths, as it underpins calculus, integration, and modelling periodic phenomena.

Why does this matter? Trigonometric equations appear in physics (wave motion, oscillations), engineering (signal processing), and even economics (seasonal trends). In exams, these questions test your algebraic manipulation, use of identities, and ability to handle multiple solutions. A common pitfall is forgetting that equations like sinθ = 0.5 have two solutions in [0°, 360°) or [0, 2π). You'll also need to handle equations involving multiples of the angle, such as sin(2θ) = 0.5, which require adjusting the interval before solving.

This topic builds on GCSE trigonometry (SOH CAH TOA, exact values) and introduces more advanced techniques like using the quadratic formula for trig equations. It's a stepping stone to solving equations with compound angles, harmonic form, and inverse trig functions. By the end, you should be able to systematically find all solutions in any given interval, using algebraic and graphical methods confidently.

Key Concepts

Core ideas you must understand for this topic

→Use the CAST diagram or unit circle to find all solutions in a given interval. For sinθ = c, solutions are θ = arcsin(c) and 180° - arcsin(c) (or π - arcsin(c) in radians). For cosθ = c, solutions are θ = arccos(c) and 360° - arccos(c) (or 2π - arccos(c)). For tanθ = c, solutions are θ = arctan(c) and 180° + arctan(c) (or π + arctan(c)).
→When solving equations like sin(2θ) = 0.5 for 0° ≤ θ ≤ 360°, first let u = 2θ, so the interval becomes 0° ≤ u ≤ 720°. Solve for u, then divide by 2 to get θ. This ensures you don't miss solutions.
→Quadratic trigonometric equations: rewrite in terms of one trig function using identities (e.g., sin²θ = 1 - cos²θ). Then solve the quadratic (e.g., 2 sin²θ - sinθ - 1 = 0 → (2 sinθ + 1)(sinθ - 1) = 0) and find all solutions for each factor.
→Always check the given interval: degrees or radians? If no interval is given, assume [0°, 360°) or [0, 2π). Use exact values (e.g., sin 30° = 1/2) where possible, but calculator use is allowed for non-exact angles.
→Be careful with domain adjustments: for tan(3θ) = 1, the interval for θ is [0°, 180°), so for 3θ it's [0°, 540°). Add 180° repeatedly to the principal solution until you exceed the upper bound.

What You Need to Demonstrate

Key skills and knowledge for this topic

Correct identification of all solutions within the specified interval
Correct use of the unit circle or graph symmetries to find multiple solutions
Correct handling of equations involving multiples of the angle (e.g., 2x) by adjusting the interval accordingly
Correct substitution and solving of quadratic equations in trigonometric functions
Correct conversion between degrees and radians when required by the question

Marking Points

Key points examiners look for in your answers

Correct identification of all solutions within the specified interval
Correct use of the unit circle or graph symmetries to find multiple solutions
Correct handling of equations involving multiples of the angle (e.g., 2x) by adjusting the interval accordingly
Correct substitution and solving of quadratic equations in trigonometric functions
Correct conversion between degrees and radians when required by the question

Examiner Tips

Expert advice for maximising your marks

💡Always check the specified interval at the start of the question
💡If solving for a multiple angle like 2x, write down the new interval for 2x before finding the principal values
💡Use a sketch of the graph or the unit circle to ensure no solutions are missed
💡Ensure your calculator is in the correct mode (degrees or radians) as specified in the question
💡For quadratic equations, treat the trigonometric function as a single variable (e.g., let u = sin x) to simplify the algebra
💡Always write down the adjusted interval when solving equations with multiples of the angle. This shows the examiner you understand the domain and reduces errors.
💡Use the CAST diagram or a quick sketch of the trig graph to verify you have the correct number of solutions. For sin and cos, there are usually two solutions per 360° period (unless at max/min).
💡If the equation is quadratic, factorise or use the quadratic formula carefully. Check that each factor gives valid solutions (e.g., sinθ = 2 has no solution). Discard any that are out of range.

Common Mistakes

Pitfalls to avoid in your exam answers

Missing solutions by failing to consider the full range of the interval
Incorrectly handling the interval when solving for multiples of the angle (e.g., failing to double the interval for 2x)
Dividing by a trigonometric function, which can lead to losing solutions
Failing to use the correct mode (degrees or radians) on the calculator
Errors in algebraic manipulation when solving quadratic trigonometric equations
Thinking sinθ = 0.5 only has one solution. Correction: sin is positive in quadrants I and II, so there are two solutions in [0°, 360°): 30° and 150° (or π/6 and 5π/6).
Forgetting to adjust the interval when the angle is multiplied. For example, solving sin(2θ) = 0.5 for 0° ≤ θ ≤ 180°: students often solve for 2θ in [0°, 180°] instead of [0°, 360°]. Correction: multiply the interval bounds by the coefficient of θ.
Confusing the solutions for tan. tanθ = 1 has solutions 45° and 225° (since tan repeats every 180°). Students often miss the second solution or add 360° incorrectly.

Frequently Asked Questions

Common questions students ask about this topic

Before You Start

Prior knowledge that will help with this topic

•Basic trigonometry: exact values of sin, cos, tan for 0°, 30°, 45°, 60°, 90° (and radians).
•Solving linear and quadratic equations algebraically.
•Understanding of the unit circle and the graphs of sin, cos, tan functions.

Key Terminology

Essential terms to know

Periodic properties and symmetry of trigonometric graphs
Algebraic reduction of quadratic trigonometric forms
Interval manipulation for multiple-angle arguments
Application of Pythagorean identities for substitution

Likely Command Words

How questions on this topic are typically asked

Solve

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Ready to test yourself?

Practice questions tailored to this topic

Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

Before You Start

Key Terminology

Likely Command Words

Ready to test yourself?

Related Topics in EDEXCEL A-Level Mathematics

Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations

Apply differentiation to find gradients, tangents and normals; maxima and minima and stationary points; points of inflection; identify where functions are increasing or decreasing

Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form

Topic Synopsis

Key Concepts & Core Principles

Exam Tips & Revision Strategies

Common Misconceptions & Mistakes to Avoid

Examiner Marking Points

Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle

Topic Overview

Key Concepts

What You Need to Demonstrate

Marking Points

Examiner Tips

Common Mistakes

Frequently Asked Questions

How do I solve sin(2x) = 0.5 for x between 0 and 360 degrees?

What is the CAST diagram and how do I use it?

How do I solve a quadratic trig equation like 2 sin²θ - sinθ - 1 = 0?

What if the equation involves tan and a multiple angle, like tan(3θ) = 1 for 0 ≤ θ ≤ 180°?

How do I handle equations like cos²θ = 1/4?

What's the difference between solving in degrees and radians?

Before You Start

Key Terminology

Likely Command Words

Ready to test yourself?

Related Topics in EDEXCEL A-Level Mathematics

Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations

Apply differentiation to find gradients, tangents and normals; maxima and minima and stationary points; points of inflection; identify where functions are increasing or decreasing

Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form