Trigonometric Functions
Class 11 · Mathematics
4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.
Degree and Radian Measure
Part of: Angles and Trigonometric Ratios
Full turns span 360 degrees or two-pi radians; one degree splits into sixty minutes of sixty seconds; radian equals arc over radius; minutes convert through sixtieths.
Why it matters. Both units size angles. Arc formula links radius to length.
Formulas
- 360 degrees = 2 pi radians, so 180 degrees = pi radians.
- Angle in radians = arc length / radius, so l = r theta.
- 1 degree = 60 minutes and 1 minute = 60 seconds.
- 1 radian is about 57 degrees 16 minutes.
Key words
- Degree
- : A 360th part of a full turn, split into 60 minutes of 60 seconds.
- Radian
- : The angle whose arc equals the radius, about 57 degrees 16 minutes.
- Arc length
- : The curve length cut by an angle, radius times angle in radians.
- Minute and second
- : The sixtieth and thirty-six hundredth parts of a degree.
Try it in two minutes
- Convert minutes to sixtieths.
- Scale by pi over 180.
- Multiply radius by angle.
Try at home. Convert your age in degrees to radians.
Trigonometric Functions and Signs
Part of: Angles and Trigonometric Ratios
Unit-circle coordinates fix sin, cos and tan as y, x and y-over-x; hypotenuse ratios bound sin and cos within minus-one to one; ASTC quadrants flip signs while period two-pi repeats values.
Why it matters. Sign rules predict values. Bounds and periods shape graphs.
Formulas
- sin(-x) = -sin x and cos(-x) = cos x.
- In quadrant II only sin stays positive; in III only tan; in IV only cos.
Key words
- Unit circle
- : Circle of radius one fixing sin, cos and tan as y, x and y over x.
- ASTC rule
- : The memory order All, Sin, Tan, Cos for positive ratios by quadrant.
- Periodic function
- : A function repeating values after a fixed interval like 2 pi.
- Tangent function
- : The ratio y over x that repeats every pi and skips odd pi/2.
Try it in two minutes
- Read coordinates on unit circle.
- Apply quadrant signs.
- Reduce by the period.
Try at home. Tabulate signs in four quadrants.
Identities and Allied Angles
Part of: Identities and Compound Angles
Sin-squared plus cos-squared anchors one; one plus tan-squared anchors sec-squared; ninety-degree shifts swap sin with cos while one-eighty flips signs and two-seventy swaps with sign flips.
Why it matters. Identities simplify expressions. Shifts reduce any angle to acute.
Formulas
- sin^2 x + cos^2 x = 1.
- 1 + tan^2 x = sec^2 x.
- 1 + cot^2 x = cosec^2 x.
- sin(pi/2 + x) = cos x and cos(pi/2 + x) = -sin x.
- sin(pi - x) = sin x and cos(pi - x) = -cos x.
- sin(pi + x) = -sin x and cos(pi + x) = -cos x.
- sin(2pi - x) = -sin x and cos(2pi - x) = cos x.
Key words
- Pythagorean identity
- : The anchor equation linking squared sine and squared cosine to one.
- Tan-secant identity
- : The secant form that follows on dividing the anchor by cos^2 x.
- Cot-cosecant identity
- : The cosecant form that follows on dividing the anchor by sin^2 x.
- Allied angle
- : Angles like pi/2 + x that swap sine with cosine up to a sign.
Try it in two minutes
- Anchor with squares-sum-one.
- Swap on ninety shifts.
- Flip on one-eighty shifts.
Try at home. Reduce 150 degrees to 30 degrees.
Sum, Double and Triple Angles
Part of: Identities and Compound Angles
Sine of sums splits into sin-cos cross terms while cosine subtracts the product pair; doubling specialises them; triple angles cube through; principal solutions bound within standard intervals.
Why it matters. Compound formulas evaluate non-standard angles. Specialisation yields fast doubles.
Formulas
- sin(x + y) = sin x cos y + cos x sin y.
- cos(x + y) = cos x cos y - sin x sin y.
- tan(x + y) = (tan x + tan y) / (1 - tan x tan y).
- sin 2x = 2 sin x cos x.
- cos 2x = cos^2 x - sin^2 x = 2 cos^2 x - 1 = 1 - 2 sin^2 x.
- tan 2x = 2 tan x / (1 - tan^2 x).
- sin x = 0 gives x = n pi; cos x = 0 gives x = (2n + 1) pi / 2.
Key words
- Sine addition rule
- : Sine of a sum splitting into cross products of sine and cosine.
- Cosine addition rule
- : Cosine of a sum subtracting the product pair.
- Double-angle form
- : The special case of sum rules when both angles are equal.
- Principal solution
- : The standard-interval answers like 0 to 2 pi for basic equations.
Try it in two minutes
- Split sums into cross terms.
- Specialise with equal angles.
- Bound principal solutions.
Try at home. Derive sin 75° from 45° + 30°.
What your child will practise
- Degree and Radian Measure: Jungle Puzzle (Clue Puzzle)
- Degree and Radian Measure: Jungle Quick Fire (Quick Fire)
- Identities and Allied Angles: Candy Puzzle (Clue Puzzle)
- Sum, Double and Triple Angles: Desert Quest (Flip and Match)
- Trigonometric Functions and Signs: Space Quick Fire (Quick Fire)
- Quiz: Degree and Radian Measure (8 questions)
- Quiz: Identities and Allied Angles (8 questions)
- Quiz: Sum, Double and Triple Angles (8 questions)
- Quiz: Trigonometric Functions and Signs (8 questions)