Advanced Trigonometry Helper

📐 10.1 Trigonometric Ratios

Trigonometry studies the relationship between sides and angles of triangles.

Right Triangle Diagram:

        A
        /|\
       / | \
      /  |  \ Hypotenuse (AO)
     /   |   \
    /    |    \
   /     |     \
  /    θ |      \
 /       |       \
O--------+--------B
    Adjacent (OB)
    
    Opposite = AB (vertical side)
    Adjacent = OB (horizontal side)
    Hypotenuse = AO (longest side)

Primary Trigonometric Ratios:

sin θ = Opposite Hypotenuse = AB AO

cos θ = Adjacent Hypotenuse = OB AO

tan θ = Opposite Adjacent = AB OB

Memory Aid: SOH CAH TOA

Sin = Opposite/Hypotenuse
Cos = Adjacent/Hypotenuse
Tan = Opposite/Adjacent

Trig Ratios on Cartesian Plane

Coordinate System:

                    y-axis
                      ↑
                      |  A(x, y)
                      | /|
                      |/r| y
                    θ |  |
    ──────────────────O──+──────→ x-axis
                         x
                         
    Where:
    • r = √(x² + y²) (distance from origin)
    • sin θ = y/r
    • cos θ = x/r  
    • tan θ = y/x

Coordinate-based Trigonometric Ratios:

sin θ = y r

cos θ = x r

tan θ = y x

Cartesian Coordinates:
r² = x² + y² (Pythagoras)

sin θ = yr

cos θ = xr

tan θ = yx

🎯 Interactive Triangle Solver

Triangle MNP with right angle at N. Angle α at M, Angle β at P.

Side MN:

Side NP:

Hypotenuse MP:

🧭 10.2 Trig Ratios in All Quadrants

CAST Rule Diagram: II | I (S)in | (A)ll -------+------- (T)an | (C)os III | IV

CAST Rule - Positive Functions by Quadrant:
• Quadrant I (0° to 90°): All positive (sin, cos, tan)
• Quadrant II (90° to 180°): Sin positive only
• Quadrant III (180° to 270°): Tan positive only
• Quadrant IV (270° to 360°): Cos positive only

🎯 Quadrant Analyzer

Enter angle (°):

Practice Questions:

Q1: If sin θ < 0 and cos θ> 0, which quadrant?
Answer: Quadrant IV (270° < θ < 360°)

Q2: If tan θ < 0 and cos θ < 0, which quadrant?
Answer: Quadrant II (90° < θ < 180°)

🔺 10.3 Solving Triangles with Trigonometry

🎯 Advanced Triangle Solver

Example: Given tan θ = -√31, where 180° < θ < 360°

Custom Problem Solver

Given cos β = p√5, where p < 0 and β ∈ [180°; 360°]

Value of p:

🧮 10.4 Using Scientific Calculator

🎯 Interactive Calculator

Angle (degrees):

Result: 0.8480

Pre-calculated Examples:

sin 58° = 0.8480

cos 222° = -0.7431

cos 238° × tan 132° =

sin²(327°)5 + tan 37° =

⭐ 10.5 Special Angles

θ	0°	30°	45°	60°	90°	180°	270°	360°
sin θ	0	12	√22	√32	1	0	-1	0
cos θ	1	√32	√22	12	0	-1	0	1
tan θ	0	√33	1	√3	Undefined	0	Undefined	0

🎯 Special Angle Lookup

Select Angle:

🔄 10.6 Reduction Formulae

Reduction Rules:
• Q2: sin(180°-θ) = sin θ; cos(180°-θ) = -cos θ; tan(180°-θ) = -tan θ
• Q3: sin(180°+θ) = -sin θ; cos(180°+θ) = -cos θ; tan(180°+θ) = tan θ
• Q4: sin(360°-θ) = -sin θ; cos(360°-θ) = cos θ; tan(360°-θ) = -tan θ
• Negative: sin(-θ) = -sin θ; cos(-θ) = cos θ; tan(-θ) = -tan θ

Co-function Identities:
sin(90° - θ) = cos θ
cos(90° - θ) = sin θ
sin(90° + θ) = cos θ
cos(90° + θ) = -sin θ

🎯 Reduction Practice

1. cos 150° = cos(180°-30°) = -cos 30° = -√32

2. sin(-45°) = -sin 45° = -√22

3. tan 480° = tan(120°) = tan(180°-60°) = -tan 60° = -√3

🔗 10.7 Fundamental Identities

Quotient Identity:
tan θ = sin θcos θ (cos θ ≠ 0)

Pythagorean Identity:
sin²θ + cos²θ = 1

Derived forms:
sin²θ = 1 - cos²θ
cos²θ = 1 - sin²θ

Example Simplifications:

Simplify: cos²θ(1 + tan²θ)
= cos²θ(1 + sin²θcos²θ)
= cos²θ + sin²θ = 1

Simplify: 1 - cos²θ1 - sin²θ
= sin²θcos²θ = tan²θ

✅ 10.8 Proving Identities

Proof Strategies:
• Start with the more complex side
• Convert all to sin and cos
• Use fundamental identities
• Find common denominators
• Factor when possible

Example Proof: sin x · tan x + cos x = 1cos x

LHS: sin x · tan x + cos x
= sin x · sin xcos x + cos x

= sin²xcos x + cos x

= sin²xcos x + cos²xcos x

= sin²x + cos²xcos x = 1cos x = RHS ✓

📐 10.9 Solving Trigonometric Equations

Solution Steps:
1. Isolate the trigonometric function
2. Find the reference angle
3. Use CAST to determine quadrants
4. Write solutions for [0°, 360°)
5. Add periodic terms for general solution

🎯 Equation Solver

Reference Angle:

General Solution:

Example: Solve 3 tan x + √3 = 0

Step 1: 3 tan x = -√3

Step 2: tan x = -√33

Step 3: Reference angle = 30°

Step 4: tan < 0 in Q2 and Q4

Step 5: x = 150° + k·180°, k ∈ ℤ

➕ 10.11 Compound & Double Angle Identities

Compound Angle Identities:
sin(α ± β) = sin α cos β ± cos α sin β
cos(α ± β) = cos α cos β ∓ sin α sin β

Double Angle Identities:
sin 2α = 2 sin α cos α
cos 2α = cos²α - sin²α = 2cos²α - 1 = 1 - 2sin²α

🎯 Example Applications

Simplify: 2 sin 15° cos 15°
= sin(2 × 15°) = sin 30° = 12

Find: sin 15°
= sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30°
= √6 - √24

⚠️ 10.12 Undefined Cases

An identity is undefined when:
• Any denominator equals zero
• tan θ appears and θ = 90° + k·180°
• Division by a trigonometric function that equals zero

Example: When is 1tan x + tan x = tan xsin²x undefined?

1. tan x = 0 (denominator in first term): x = k·180°

2. sin²x = 0 (denominator in RHS): x = k·180°

3. tan x undefined: x = 90° + k·180°

Combined: x = k·90°, k ∈ ℤ

🔺 Advanced Trigonometry Helper