Trigonometric Ratios — Values, signs in quadrants
Trigonometric Functions & Equations
8
JEE Qs
8%
Hard
60
min
Master the unit circle and the ASTC rule for rapidly determining signs and values of trigonometric ratios for any angle, crucial for speed and accuracy in problems.
🧮 Key Formulas
✅ Key Points for JEE
- 1The signs of trigonometric ratios in different quadrants are determined by the 'All Silver Tea Cups' (or ASTC) rule: All positive in Q1, Sin positive in Q2, Tan positive in Q3, Cos positive in Q4.
- 2For angles greater than 90 degrees or negative angles, first identify the quadrant to determine the sign, then use the reference angle (acute angle formed with the x-axis) for the magnitude of the ratio.
- 3Master the exact values of trigonometric ratios for standard angles (0°, 30°, 45°, 60°, 90°) and their radian equivalents (0, pi/6, pi/4, pi/3, pi/2).
- 4Visualize trigonometric ratios using the unit circle where for any point (x,y) on the circle, x=cos(theta) and y=sin(theta), providing a direct way to see signs and periodic nature.
- 5Remember the reciprocal relationships: cosec(theta) = 1/sin(theta), sec(theta) = 1/cos(theta), cot(theta) = 1/tan(theta).
⚠️ Common Mistakes
- ✕Incorrectly assigning signs to trigonometric ratios in various quadrants, especially when dealing with angles outside [0, 90°] or negative angles.
- ✕Mixing up reciprocal identities (e.g., confusing sec(theta) with cosec(theta) or cot(theta) with tan(theta)).
- ✕Failing to recall or incorrectly recalling the exact values of trigonometric ratios for standard angles (0, 30, 45, 60, 90 degrees/radians).
- ✕Algebraic errors when simplifying expressions involving squares of ratios (e.g., writing sin(theta) + cos(theta) = 1 instead of sin^2(theta) + cos^2(theta) = 1).
📝 Practice Questions
See allQ11.Let the range of the function f(x) = 6 + 16 cos x ⋅cos ( π3 −x) ⋅cos ( π3 + x) ⋅sin 3x ⋅cos 6x, x ∈R be [α, β] . Then the distance of the point (α, β) from the line 3x + 4y + 12 = 0 is : (1) 11 (2) 8 (3) 10 (4) 9 sin y > 0 and x(1) = π2 . Then
Q7. If ∑13r=1 { sin( 4 +(r−1) 6 ) sin( π4 + rπ6 ) } (1) 10 (2) 4 (3) 2 (4) 8
Q18.The sum of all values of θ ∈[0, 2π] satisfying 2 sin2 θ = cos 2θ and 2 cos2 θ = 3 sin θ is 2025 (22 Jan Shift 2) JEE Main Previous Year Paper (1) 4π (2) 5π6 (3) π (4) π 2
Q18.The value of (sin 70∘) (cot 10∘cot 70∘−1) is (1) 2/3 (2) 1 (3) 0 (4) 3/2 dx 1 1 1 , then 3( b + c) is equal to
Q22.If for some α, β; α ≤β, α + β −8 and sec2 (tan−1 α) + cosec2 (cot−1 β) −36, then α2 + β is_______. Q23. ⎡x⎤ Let A be a 3 × 3 matrix such that X TAX = O for all nonzero 3 × 1 matrices X = y . If ⎣z ⎦ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡1 ⎤ ⎡ 0 ⎤ A 1 = 4 , A 2 = 4 , and det(adj(2(A + 1))) −2α3β5γ, α, β, γ ∈N , then α2 + β2 + γ 2 ⎣ 1⎦ ⎣ −5 ⎦ ⎣1⎦ ⎣−8 ⎦ is_____. x ≥0. Then
Q20.If sin x + sin2 x = 1, x ∈(0, π2 ), then (cos12 x + tan12 x) + 3 (cos10 x + tan10 x + cos8 x + tan8 x) + (cos6 x + tan6 x) is equal to : (1) 4 (2) 1 (3) 3 (4) 2 π
NCERT Chapters
- Class 10 Mathematics Ch 8: Introduction to Trigonometry
- Class 11 Mathematics Ch 3: Trigonometric Functions