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CIE A-Level Maths Study Notes

2.3.2 Trigonometric Identities and Simplification

Trigonometric equations, integral to A-Level Pure Mathematics, involve the manipulation and solution of equations containing trigonometric functions. These equations are pivotal in understanding various mathematical phenomena, particularly in fields like physics, engineering, and geometry. This section aims to provide a comprehensive guide to solving trigonometric equations, tailored for A-Level students.

Understanding and Applying Trigonometric Identities

Trigonometric identities are invaluable tools in mathematics for simplifying and evaluating complex expressions. Mastery of these identities is essential for solving a variety of trigonometry problems. This section covers key identities along with solutions to example problems for clear understanding.

Basic Trigonometric Identities

Pythagorean Identities

1. Identity: (cosθ)2+(sinθ)21(\cos \theta)^2 + (\sin \theta)^2 \equiv 1

  • Example: If sinθ=35\sin \theta = \frac{3}{5}, find cosθ\cos \theta.
  • Solution: cosθ=±1(35)2=±0.8\cos \theta = \pm \sqrt{1 - \left(\frac{3}{5}\right)^2} = \pm 0.8

2. Identity: 1+(tanθ)2(secθ)21 + (\tan \theta)^2 \equiv (\sec \theta)^2

  • Example: Simplify 1+tan245.1 + \tan^2 45^\circ.
  • Solution: 1+1=21 + 1 = 2

3. Identity: (cotθ)2+1(cosecθ)2(\cot \theta)^2 + 1 \equiv (\cosec \theta)^2

  • Example: Verify cot245+1=cosec245\cot^2 45^\circ + 1 = \cosec^2 45^\circ.
  • Solution: (11)2+1=(21)2\left(\frac{1}{1}\right)^2+1=\left(\frac{\sqrt{2}}{1}\right)^2, which simplifies to 2=22=2, verifying the identity.

Double Angle Identities

1. Sin Double Angle

  • Identity: sin2A2sinAcosA\sin 2A \equiv 2 \sin A \cos A
    • Example: Find sin2A\sin 2A when sinA=12\sin A = \frac{1}{2} and cosA=32.\cos A = \frac{\sqrt{3}}{2}.
      • Solution: sin2A=2×12×32=32\sin 2A = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}

2. Cos Double Angle

  • Identity: cos2A(cosA)2(sinA)2\cos 2A \equiv (\cos A)^2 - (\sin A)^2
    • Example: Express cos60\cos 60^\circ using the double angle identity.
      • Solution: cos60=2×(12)21=0.\cos 60^\circ = 2 \times \left(\frac{1}{2}\right)^2 - 1 = 0.

3. Tan Double Angle

  • Identity: tan2A2tanA1(tanA)2\tan 2A \equiv \frac{2 \tan A}{1 - (\tan A)^2}
    • Example: Simplify tan2A\tan 2A when tanA=1\tan A = 1.
      • Solution: Undefined (due to division by zero)

Addition and Subtraction Identities

1. Sin Addition and Subtraction

  • Identity: sin(A±B)sinAcosB±cosAsinB\sin (A \pm B) \equiv \sin A \cos B \pm \cos A \sin B
    • Example: Find sin(45+30)\sin (45^\circ + 30^\circ).
      • Solution: sin75=24+640.966\sin 75^\circ = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} \approx 0.966

2. Cos Addition and Subtraction

  • Identity: cos(A±B)cosAcosBsinAsinB\cos (A \pm B) \equiv \cos A \cos B \mp \sin A \sin B
    • Example: Calculate cos(6045)\cos (60^\circ - 45^\circ).
      • Solution: cos15=24+640.966\cos 15^\circ = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} \approx 0.966

3. Tan Addition and Subtraction

  • Identity: tan(A±B)tanA±tanB1tanAtanB\tan (A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}
    • Example: Simplify tan(45+30)\tan (45^\circ + 30^\circ).
      • Solution: tan75=3/3+113/33.732\tan 75^\circ = \frac{\sqrt{3}/3 + 1}{1 - \sqrt{3}/3} \approx 3.732
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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