Trigonometric Equations with Multiple Angles
The diagram shows where the various ratios are positive Note also that values repeat every 360o 90° 180 - S A 180° 0°,360° C 180 + T 360 - 270°
Solve cos 3x = – 1/√2 for values of x such that 0 < x < π First of all consider the range of values If 0 < x < π then 0 < 3x < 3π (540o) Remember 3π = 3 × 180o = 540o Work in degrees then change back to radians later Now find BASE angle cos -1 (1/√2 ) = 45o π/4 A S T C 180 – Since cosine negative 3x = 135o , 225o 495o 180 + Now change back to radians using multiples of π/4 3x = 3π/4 , 5π/4 , 11π/4 So x = 3π/12 , 5π/12 , 11π/12
Solve sin 2x = 1/2 for values of x such that 0 < x < 2π First of all consider the range of values If 0 < x < 2π then 0 < 2x < 4π (720o) Work in degrees then change back to radians later Now find BASE angle Remember 4π = 4 × 180o = 720o sin -1 (1/2 ) = 30o π/6 A S T C 180 – Since sine positive 2x = 30o , 150o 390o , 510o Now change back to radians using multiples of π/6 2x = π/6 , 5π/6 , 13π/6 ,17π/6 So x = π/12 , 5π/12 , 13π/12 , 17π/12
Solve tan 4x = √3 for values of x such that 0 < x < π First of all consider the range of values If 0 < x < π then 0 < 4x < 4π (720o) Remember 4π = 4 × 180o = 720o Now find BASE angle Work in degrees then change back to radians later tan -1 (√3 ) = 60o π/3 A S T C Since tangent positive 180 + 4x = 60o , 240o 420o , 600o Now change back to radians using multiples of π/3 4x = π/3 , 4π/3 , 7π/3 ,10π/3 So x = π/12 , π/3 , 7π/12 , 5π/6
Solve 2sin (2x – π/6) = √3 2x – π/6 = 60o , 120o for values of x such that 0 < x < 2π First of all consider the range of values If 0 < x < 2π then 0 < 2x – π/6 < 4π – π/6) (690o) Remember 2π – π/6 = 2 × 180o – 30o = 690o Now find BASE angle Work in degrees then change back to radians later A S T C sin -1 (√3/2 ) = 60o π/3 180 - 2x – π/6 = 60o , 120o 420o , 480o Now change back to radians using multiples of π/3 2x – π/6 = π/3 , 2π/3 , 7π/3 ,8π/3 So x = π/4 , 5π/12 , 5π/6 , 17π/12 2x = 3π/6 , 5π/6 , 15π/6 ,17π/6