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

The idea

This topic requires solving trigonometric equations within a specified interval, such as 0° ≤ θ < 360° or 0 ≤ x < 2π. You must handle simple equations like sin θ = k, quadratic equations in sin, cos or tan (e.g., 2sin²θ - sinθ - 1 = 0), and equations with multiples of the angle (e.g., sin 2θ = 0.5). The method involves finding the principal value, using symmetry of trigonometric graphs to find all solutions in the interval, and adjusting for multiples by dividing the interval accordingly. For quadratic equations, factorise or use the quadratic formula, then solve each factor. Always check solutions are within the given interval and consider degrees or radians as specified.

What earns the marks

• Find the principal value using inverse trigonometric functions, considering the correct range.
• Use the symmetry and periodicity of trigonometric functions to find all solutions in the given interval.
• Solve quadratic equations in sin θ, cos θ or tan θ by factorising or using the quadratic formula.
• Handle equations involving multiples of the angle, such as sin 2θ = k, by adjusting the interval for 2θ and then dividing by the multiple.
• Reject solutions outside the specified interval and present answers clearly.

Common mistakes and how to put them right

• Forgetting to find all solutions in the interval, only giving the principal value.

  Put it right: Use the unit circle or graphs to find additional solutions.

• Incorrectly factorising quadratic trigonometric equations.

  Put it right: Treat sin θ as a variable and factorise carefully, or use the quadratic formula.

• Not adjusting the interval when solving equations with multiples of the angle.

  Put it right: If solving sin 2θ = k for 0° ≤ θ < 360°, first solve for 2θ in 0° ≤ 2θ < 720°, then divide by 2.

Exam tips

• Always check whether the interval is in degrees or radians and work consistently.
• Sketch the trigonometric function to visualise the number of solutions and their approximate locations.
• For quadratic equations, ensure you consider all possible solutions from each factor, including those that may be invalid (e.g., sin θ = 2 has no solution).