1) Introduction to Trigonometry

Trigonometry studies relationships between angles and sides in triangles. The skills here power real‑world fields such as architecture, satellites, mobile networks, and even finance analytics where periodic behavior (waves) matters. Think of trig as a language for measuring turn (angle) and length.

2) Right‑Angled Triangles & Key Terms

• Hypotenuse (H): side opposite the right angle; longest side.
• Opposite (O): side opposite the angle θ.
• Adjacent (A): side next to θ, not the hypotenuse.

Label the triangle relative to the angle you are working with; the same triangle can have different O and A for different angles.

3) Trigonometric Ratios (SOH‑CAH‑TOA)

Memory aid: SOH‑CAH‑TOA.

Example: In a right triangle, θ faces a side of 6 and the hypotenuse is 10. Then sin θ = 6/10 = 3/5. Use a calculator in degree mode to find θ if needed.

4) Reciprocal Ratios

• csc θ = 1/sin θ = H/O
• sec θ = 1/cos θ = H/A
• cot θ = 1/tan θ = A/O

Knowing reciprocals helps with algebraic simplification and exam manipulation questions.

5) Special Angles (0°, 30°, 45°, 60°, 90°)

Build‑from‑geometry: Use 30‑60‑90 and 45‑45‑90 triangles to derive these values—great for mental math in tests.

6) Pythagoras Refresher

For right triangles: H² = A² + O². Often you first use Pythagoras to find a missing side, then apply a ratio. This is foundational for STEM careers like civil engineering and computer graphics.

Example: Legs 5 and 12 ⇒ H = 13.

7) Core Identities

• sin² θ + cos² θ = 1
• 1 + tan² θ = sec² θ
• 1 + cot² θ = csc² θ

These always hold and allow you to swap between functions. They appear frequently in exam preparation questions.

8) Simplifying Trig Expressions

Techniques: convert all terms to sin and cos, use reciprocal and quotient identities, factor common terms, and use sin²+cos²=1 to reduce powers.

Example: Simplify sec² θ − tan² θ ⇒ using 1 + tan² θ = sec² θ gives 1.

9) Unit Circle & Cartesian Plane

Place a radius‑1 circle at the origin. A point at angle θ has coordinates (cos θ, sin θ). This links graphs, identities, and signs across quadrants and is essential for advanced math tutoring and online courses.

10) Reduction Formulae (ASTC)

All Students Take Calculus: In Quadrants I‑IV, the positive functions are All, Sin, Tan, Cos respectively.

11) Compound Angle Formulae

• sin(A±B) = sin A cos B ± cos A sin B
• cos(A±B) = cos A cos B ∓ sin A sin B
• tan(A±B) = (tan A ± tan B)/(1 ∓ tan A tan B)

Example: Find sin 75° ⇒ sin(45°+30°) = √2/2·1/2 + √2/2·√3/2 = (√2+√6)/4.

12) Double Angle Formulae

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ
• tan 2θ = (2 tan θ)/(1 − tan² θ)

These are powerful for converting between powers and angles in simplification and integration later on.

13) Solving Basic Trig Equations

1. Isolate the trig function.
2. Find the reference angle using inverse trig.
3. Use ASTC to get all angles in the interval.

Example: sin θ = 1/2 on [0°,360°] ⇒ θ = 30°, 150°.

14) General Solutions

We express infinite solutions using period:

• sin θ = k ⇒ θ = α + 360°n or θ = (180°−α) + 360°n.
• cos θ = k ⇒ θ = ±α + 360°n.
• tan θ = k ⇒ θ = α + 180°n.

Here α is a principal angle and n ∈ ℤ. This format is common in senior grades and standardized tests for university admissions and scholarship applications.

15) Graphs: Sine, Cosine, Tangent

• y = sin x and y = cos x have range [−1, 1] and period 360°.
• y = tan x has vertical asymptotes at 90° + 180°k and period 180°.

Sketch key points every 90° (or 45°) to draw quick, accurate graphs during exams.

16) Transformations of Graphs

General form y = a · f(bx + c) + d.

• Amplitude = |a| (height of sine/cosine).
• Period = 360°/|b| for sin/cos; 180°/|b| for tan.
• Phase shift = −c/b; vertical shift = d.

Example: y = 2 sin(3x − 60°) − 1 ⇒ amplitude 2, period 120°, phase shift 20°, midline y = −1.

17) Heights & Distances (Applications)

Use angles of elevation/depression with SOH‑CAH‑TOA. These skills connect to surveying, drone mapping, and property valuation problems that also appear in real‑estate analytics.

Example: A 12 m shadow at 35° elevation ⇒ height = 12 · tan 35°.

18) Bearings & Navigation

Bearings are measured clockwise from North (000° to 360°). Convert between bearing angles and Cartesian angles for calculations. Useful in aviation, shipping, and GPS—topics often discussed in technology and insurance risk assessment.

19) Physics & Engineering Links

• Resolve forces: Fx = F cos θ, Fy = F sin θ.
• Waves & electricity: sinusoidal models for AC circuits.
• Rotational motion: angles in radians; arc length s = rθ.

These connections support career paths in engineering, software development, and data science.

20) Study Tips & Exam Strategy

Extra: If you’re exploring scholarships, student finance, or credit score basics for university readiness, pair these notes with financial‑literacy guides. Math + money skills = powerful combo.