### Trigonometry

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→ –1 ≤ sinθ, cosθ ≤ + 1

→ – α ≤ tanθ, cotθ ≤ + α

→ secθ, cosecθ ≥ 1, or secθ, cosecθ ≤ – 1

→ sinθ. cosecθ = cosθ. secθ = tanθ. cotθ = 1

→ sin2θ + cos2θ = 1

→ sec2θ – tan2θ = 1

→ cosec2θ – cot2θ = 1

→ Maximum value of  m sinθ ± n cosθ =

sin1°. sin2°. sin3°. ............. sin 180° = 0

→ cos1°. cos2°. cos3°............... cos90° = 0

→ tan1°. tan2°. tan3°................. tan89° = 1

→ cot1°. cot2°. cot3°............. cot89° = 1

→ If secθ ± tanθ = x , then secθ =

→ If sinθ + cosθ = x, then sinθ – cosθ =

→ Angle and Measurement

→ (90 ± θ), (270 ± θ) ⇒ sinθ → cosθ

cosθ → sinθ

tanθθ ↔ cot θ

secθθ ↔ cosecθ

→ Compound Angle

→ sin (A + B) = sinA. cosB + cosA. sinB

→ sin (A – B) = sinA. cosB – cosA . sinB

→ cos (A + B) = cosA. cosB – sinA. sinB

→ cos (A – B) = cosA. cosB + sinA. sinB

→ tan (A ± B) =

→ sin2A = 2sinA cosA = = 2cos2A –1

→ cos2A = cos2A - sin2A =

→ tan2A =

→ sin3A = 3 sinA – 4 sin3A

→ cos3A = 4cos3A – 3 cosA

→ tan3A =

→ sinC + sinD = 2sin . cos

→ sinC – sinD = 2 sin . cos

→ cosC + cosD = 2cos . cos

→ cosC – cosD = 2sin . sin

= R (R → Circumradius of triangle)

→ Cosine Rule → cosA =

→ cosB =

→ cosC =

→ sinθ. sin2θ . sin4θ =

→ cosθ. cos2θ. cos4θ =

→ tanθ. tan2θ. tan4θ = tan3θ

→ sin18° = cos 72° =

→ cos 18° = sin72° =

→ sin36° = cos54° =

→ cos36° = sin54° =

→ sin15° = cos75° =

→ cos 15° = sin75° =

→ tan15° =

→ Measurement of angle

i) degree system :- 1° = 1 degree = right angle

1 minute  = 11 = degree

111 = 1second= minute

ii) Circular system :- unit → Radian

1° = 1 radian = 57°16'22'' approx.

→ The angle between two consecutive digits in a clock is 30°

→ The hour hand rotates through an angle of 30° in one hour i.e in one minute.

→ The minute hand rotates through an angle of 6° in one minute.

→ If a sinθ + b cosθ = c and c2 = a2+b2, then

Base = b, perpendicular = a, hypotenuse = c

→ Maximum value of sin4θ.cos4θ is

→ Minimum value of : a sec2θ + b cosec2θ =

→ o ≤ sin2 θ ≤ 1 , 0 ≤ cos2θ ≤

→ sin2nθ + cos2mθ ≤ 1

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