Trigonometry is the language of angles, waves, and rotational motion. From architecture and astronomy to AI and signal processing, it has powerful real-world applications. Whether you’re preparing for A-Level Maths, IB Math AA HL, or AP Precalculus, this guide will walk you through every major concept with clarity and purpose.

🧭 Why Trigonometry Matters

Trigonometry connects algebra and geometry in a deep and beautiful way. It teaches you to:

• Understand periodic behavior
• Model real-world phenomena like sound, motion, or light
• Solve complex geometry and engineering problems

If you’re aiming for a 7 in IB, A* in A-Level, or 5 in AP, mastering trigonometry is non-negotiable.

📐 1. Trigonometric Ratios

At its core, trigonometry is about the relationships between sides and angles in triangles.

For a right-angled triangle:

• sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}sinθ=hypotenuseopposite​
• cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}cosθ=hypotenuseadjacent​
• tan⁡θ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}}tanθ=adjacentopposite​

These are SOH-CAH-TOA — essential for problem-solving in triangles and wave applications.

🔄 2. Inverse Trigonometric Functions

Inverse functions help us find angles when side ratios are known.

• sin⁡−1(x),cos⁡−1(x),tan⁡−1(x)\sin^{-1}(x), \cos^{-1}(x), \tan^{-1}(x)sin−1(x),cos−1(x),tan−1(x)

They are especially important in geometry, solving equations, and calculus integrals.

📏 3. Solving Triangles: Right and Non-Right

🔹 Right Triangles:

Use basic trig ratios and the Pythagorean theorem.

🔹 Non-Right Triangles:

Use:

• Sine Rule: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}sinAa​=sinBb​=sinCc​
• Cosine Rule: c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 – 2ab\cos Cc2=a2+b2−2abcosC
• Area Rule: 12absin⁡C\frac{1}{2}ab\sin C21​absinC

Crucial for IB AA HL Paper 2, A-Level Pure Math, and AP Precalculus.

🔄 4. Unit Circle and Radians

The unit circle helps extend trigonometry beyond right triangles.

• Radius = 1
• Coordinates = (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta)(cosθ,sinθ)

Radians are the natural angle measure in higher mathematics:

• 180∘=π180^\circ = \pi180∘=π radians

All advanced trigonometric analysis uses radians.

🌊 5. Graphs of Sine, Cosine, and Tangent

Understanding graphs is critical in physics, signal processing, and calculus.

• Sine: periodic with period 2π2\pi2π
• Cosine: similar to sine, but starts at max
• Tangent: vertical asymptotes, period π\piπ

Key features:

• Amplitude
• Period
• Phase shift
• Vertical shift

Graph transformations are heavily tested in A-Level and IB HL exams.

🔗 6. Trigonometric Identities

These are true for all values in their domain and help simplify expressions:

• sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1
• 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta1+tan2θ=sec2θ
• cot⁡2θ+1=csc⁡2θ\cot^2 \theta + 1 = \csc^2 \thetacot2θ+1=csc2θ

Used in integration, solving trig equations, and calculus.

➕ 7. Double Angle and Compound Angle Formulas

Used in simplifying expressions and advanced proofs:

• sin⁡(2x)=2sin⁡xcos⁡x\sin(2x) = 2\sin x \cos xsin(2x)=2sinxcosx
• cos⁡(2x)=cos⁡2x−sin⁡2x\cos(2x) = \cos^2 x – \sin^2 xcos(2x)=cos2x−sin2x
• sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin Bsin(A±B)=sinAcosB±cosAsinB
• cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin Bcos(A±B)=cosAcosB∓sinAsinB

These formulas appear frequently in IB HL Paper 3 and A-Level Mechanics.

🧮 8. Trigonometric Equations

You’ll often solve equations like: sin⁡x=12,tan⁡2x=3\sin x = \frac{1}{2}, \quad \tan 2x = \sqrt{3}sinx=21​,tan2x=3​

Steps usually involve:

• Using identities
• Restricting domain (especially in radians)
• Solving general and specific solutions

Understanding periodicity is key to finding all solutions.

📐 9. Applications in Geometry and Modeling

Trigonometry powers real-world applications like:

• Architecture and Engineering
• Modeling sound, light, or periodic motion
• Projectile and circular motion in physics
• Fourier Analysis (advanced)

You’ll also apply trigonometry in vector problems, complex numbers (in polar form), and even 3D geometry.

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