Trigonometry - MedhaVatika

Trigonometry Adventure

Step into an exciting journey where triangles come alive and every angle tells a story! Trigonometry Adventure by MedhāVatika transforms one of the most challenging math topics into a playful exploration.

Chapter 1 Degrees and Radians

Degrees (°): A common unit for measuring angles. A full circle is divided into 360 degrees.
Radians (rad): A mathematical unit for measuring angles based on the radius of a circle. A full circle is 2π radians.
Conversion Formulas:
- To convert degrees to radians: angle in degrees × (π / 180°)
- To convert radians to degrees: angle in radians × (180° / π)

Solutions to Exercise 1:

How many degrees are in π radians?
- Since 2π radians = 360°, then π radians is half of that.
- Answer: 180°
Convert 90° to radians.
- Using the formula: 90° × (π / 180°) = 90π / 180 = π/2 radians.
- Answer: π/2 radians
Convert π/3 radians to degrees.
- Using the formula: (π/3) × (180° / π) = 180°/3 = 60°.
- Answer: 60°

Degrees ↔ Radians — Super Simple

Degrees ↔ Radians — Unit Circle

Drag/click the circle or use the slider to set θ

Radians: 0.000

Degrees: 0.0°

Exact: —

x = cosθ: 1.000

y = sinθ: 0.000

Chapter 2 The Six Trig Functions

SOH CAH TOA is a mnemonic for the three primary trigonometric functions in a right-angled triangle.

- - Sine (sin) = Opposite / Hypotenuse
  - Cosine (cos) = Adjacent / Hypotenuse
  - Tangent (tan) = Opposite / Adjacent

Reciprocal Functions: Each primary function has a reciprocal.

- - Cosecant (csc) = 1/sin = Hypotenuse / Opposite
  - Secant (sec) = 1/cos = Hypotenuse / Adjacent
  - Cotangent (cot) = 1/tan = Adjacent / Opposite

Six Trig Functions — Standalone Simulator

Six Trig Functions — SOH CAH TOA

Primary and reciprocal functions in a right triangle

Angle: 0°

Hypotenuse: 1.000

Opposite: 0.000

Adjacent: 1.000

sin

csc

—

cos

sec

tan

cot

—

Drag the slider or click the circle to set the angle. Values update for primary and reciprocal functions.

How Sin²θ + Cos²θ = 1?

sin²θ + cos²θ = 1 — Interactive Mini Simulator

sin(θ) cos(θ)

sinθ = 0.0000
sin²θ = 0.0000

cosθ = 1.0000
cos²θ = 1.0000

sin²θ + cos²θ = 1.0000 → identity holds ✔ (pointwise)

cos²θ 100.0%sin²θ 0.0%

Tip: drag the cyan point, use the slider, or press Play. The circle, waves, shaded integrals, and the 100% bar all stay in sync.

Chapter 4: Shape the wave

How does its period compare to a regular cosine wave? Is it shorter or longer?

Shape the Wave

Amplitude A: 1.0

Frequency B: 1.0

Phase φ (deg): 0°

Vertical shift D: 0.0

Target

Your amplitude: 1.00 Your period: 6.283 Target amplitude: 3 Target period: π ≈ 3.142 Score: — Match the target!

Practical Applications

Navigation & mapping

GPS & geodesy: Convert satellite ranges to latitude/longitude using spherical trig; local ENU ↔ geodetic conversions.

Triangulation: Find your position by measuring angles to two known landmarks.

Aviation & marine: Course corrections with wind/current using vector components: $vy=vsin⁡θv_x=v\cos\theta,\ v_y=v\sin\theta$ .

Surveying, civil & architecture

Measuring heights/distances: Clinometer + baseline: $\text{height} \approx d\tan\theta + \text{eye height}$ .

Road & rail design: Grades, curves, and superelevation (bank angle) from $⁡θ=v2/(rg)\tan\theta = v^2/(rg)$ .

Roofs & stairs: Pitch and rise–run angles; stringer length via cosine rule.

Physics of motion (sports & ballistics)

Projectile motion: Range $R=v2gsin⁡2θR=\frac{v^2}{g}\sin 2\theta$ ; optimal launch angles for javelin, basketball arcs, free-kicks.

Spin & curve balls: Magnus effect depends on spin axis angle; components set with trig.

Golf/archery analytics: Decompose club/bow speed into launch speed & direction.