sinθ space equals space straight p over straight h space comma space cotθ space equals space straight b over straight p

cosθ space equals space straight b over straight h comma space secθ space equals straight h over straight b

tanθ space equals straight p over straight b comma cosecθ space equals space straight h over straight p

→ –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 = square root of straight m squared plus straight n squared end root

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 = open parentheses fraction numerator straight x squared plus 1 over denominator 2 straight x end fraction close parentheses

→ If sinθ + cosθ = x, then sinθ – cosθ = square root of 2 minus straight x squared end root

→ 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) = open parentheses fraction numerator tanA space plus-or-minus space tanB over denominator 1 minus-or-plus tanA. space tanB end fraction close parentheses

→ sin2A = 2sinA cosA = open parentheses fraction numerator 2 tanA over denominator 1 plus tan squared straight A end fraction close parentheses = 2cos2A –1

→ cos2A = cos2A - sin2A = open parentheses fraction numerator 1 – tan squared straight A over denominator 1 plus tan squared straight A end fraction close parentheses equals 2 cos squared straight A space minus 1

→ tan2A = open parentheses fraction numerator 2 tanA over denominator 1 minus tan squared straight A end fraction close parentheses

→ sin3A = 3 sinA – 4 sin3A

→ cos3A = 4cos3A – 3 cosA

→ tan3A = open parentheses fraction numerator 3 tanA – 3 tan cubed straight A over denominator 1 plus 3 tan squared straight A end fraction close parentheses

→ sinC + sinD = 2sin open parentheses fraction numerator straight C plus straight D over denominator 2 end fraction close parentheses. cos  open parentheses fraction numerator straight C – straight D over denominator 2 end fraction close parentheses

→ sinC – sinD = 2 sin open parentheses fraction numerator straight C – straight D over denominator 2 end fraction close parentheses. cos open parentheses fraction numerator straight C plus straight D over denominator 2 end fraction close parentheses

→ cosC + cosD = 2cos open parentheses fraction numerator straight C plus straight D over denominator 2 end fraction close parentheses. cos open parentheses fraction numerator straight C – straight D over denominator 2 end fraction close parentheses

→ cosC – cosD = 2sin open parentheses fraction numerator straight C plus straight D over denominator 2 end fraction close parentheses. sin open parentheses fraction numerator straight D – straight C over denominator 2 end fraction close parentheses

straight a over sinA equals straight b over sinB equals straight c over sinB= R (R → Circumradius of triangle)

→ Cosine Rule → cosA = open parentheses fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction close parentheses

                         → cosB = open parentheses fraction numerator straight a squared plus straight c squared minus straight b squared over denominator 2 ac end fraction close parentheses

                         → cosC = open parentheses fraction numerator straight a squared plus straight b squared minus straight c squared over denominator 2 ab end fraction close parentheses

→ sinθ. sin2θ . sin4θ = 1 fourth sin 3 straight theta

→ cosθ. cos2θ. cos4θ = 1 fourth cos 3 straight theta

→ tanθ. tan2θ. tan4θ = tan3θ

→ sin18° = cos 72° = open parentheses fraction numerator square root of 5 minus 1 over denominator 4 end fraction close parentheses

→ cos 18° = sin72° = open parentheses fraction numerator square root of 10 plus 2 square root of 5 end root over denominator 4 end fraction close parentheses

→ sin36° = cos54° = open parentheses fraction numerator square root of 10 – 2 square root of 5 end root over denominator 4 end fraction close parentheses

→ cos36° = sin54° = open parentheses fraction numerator square root of 5 plus 1 over denominator 4 end fraction close parentheses

→ sin15° = cos75° = open parentheses fraction numerator square root of 3 minus 1 over denominator 2 square root of 2 end fraction close parentheses

→ cos 15° = sin75° = open parentheses fraction numerator square root of 3 plus 1 over denominator 2 square root of 2 end fraction close parentheses

→ tan15° = open parentheses fraction numerator square root of 3 minus 1 over denominator square root of 3 plus 1 end fraction close parentheses

→ Measurement of angle

i) degree system :- 1° = 1 degree = 1 over 90 right angle

                                              1 minute  = 11 = 1 over 60degree

                                              111 = 1second= 1 over 60 minute

 

ii) Circular system :- unit → Radian

1° = straight pi over 180radian = open parentheses fraction numerator 22 over denominator 7 cross times 180 end fraction close parentheses radian

1° = 1 radian = 57°16'22'' approx.

1° = 0.01746 radian

→ The angle between two consecutive digits in a clock is 30° open parentheses equals straight pi over 6 radians close parentheses

→ The hour hand rotates through an angle of 30° in one hour i.e open parentheses 1 half close parentheses 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 open parentheses 1 half close parentheses to the power of straight n

→ Minimum value of : a sec2θ + b cosec2θ = open parentheses square root of straight a plus square root of straight b close parentheses squared

→ o ≤ sin2 θ ≤ 1 , 0 ≤ cos2θ ≤

→ sin2nθ + cos2mθ ≤ 1

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