Trigonometric Functions — Class 12 Maharashtra HSC Mathematics revision notes
A 7-page visual revision sheet for Trigonometric Functions, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Trigonometric Functions in one look
equations that repeat forever, triangles solved from one side, and functions run backward
Trigonometric Equations
- an equation in trigonometric ratios of an unknown angle: e.g. sinθ=1/2, tanθ=2, cos3θ=cos5θ
- any value of θ satisfying the equation is a solution
- every trig ratio repeats (sin, cos every 2π, tan every π), so infinitely many solutions can exist
- some equations have no solution at all: sinθ=3 is impossible, since sinθ never leaves [-1,1]
Principal Solutions
- the interval searched, one full revolution
0≤θ<2π - worked: cosθ=1/2 has two principal solutions
θ=π/3, 5π/3
General Solution
- sinθ=sinα
θ=nπ+(-1)^nα - cosθ=cosα
θ=2nπ±α - tanθ=tanα
θ=nπ+α - sinθ=0
θ=nπ - cosθ=0
θ=(2n+1)π/2 - tanθ=0
θ=nπ - squared sin, cos or tan equation
θ=nπ±α
Page 1 of 7. The full sheet, all 7 pages, opens free in the app.
What this Trigonometric Functions sheet covers
- Trigonometric Equations
- Principal Solutions
- the interval searched, one full revolution
- worked: cosθ=1/2 has two principal solutions
- General Solution
- sinθ=sinα
- cosθ=cosα
- tanθ=tanα
- sinθ=0
- cosθ=0
- tanθ=0
- squared sin, cos or tan equation
- General Solution of tan theta = 1
- Polar Co-ordinates
- Sine Rule
- in any triangle ABC, R is the circumradius
- area, from two sides and the angle between them
- Cosine Rule
- solved for the side
- solved for the angle
- Area of a Triangle, Three Ways
- two sides and the included angle
- all three sides and the circumradius
- all three sides, Heron's formula
- Projection Rule
- a side as the sum of the other two sides' projections
- the same pattern for the other two sides
- Half Angle Formulas, an application
- sinA/2
- cosA/2
- tanA/2
- area, Heron's formula
- Napier's analogy
- Inverse Trigonometric Functions
- Principal Values
- sin^-1x
- cos^-1x
- tan^-1x
- cot^-1x
- sec^-1x
- cosec^-1x
- Special Values: Inverse Trig at 0, 1, -1
- Which Rule to Reach For
- Sine Rule fits
- Cosine Rule fits
- Worked Principal Values
- Co-function, Reciprocal and Sum-Difference Properties
- co-function pairs sum to a right angle
- reciprocal relation, needs |x|≥1
- reciprocal relation for tan inverse, sign dependent
- sin inverse, odd
- tan inverse, odd
- cosec inverse, odd
- cos inverse, pi complement
- sec inverse, pi complement
- cot inverse, pi complement
- function then inverse, holds on the whole domain
- inverse then function, needs y on the principal branch
- sum, safe zone xy<1
- difference, needs xy>-1
- sum correction, x,y>0 and xy>1
- sum correction, x,y<0 and xy>1
- sum, x,y>0 and xy=1
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Trigonometric Functions the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.