To find the values of the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) for the angles 120°, 135°, 150°, 180°, and 270°, we rely on the unit circle and the properties of these functions.
Angle 120°
- sin(120°) = √3/2
- cos(120°) = -1/2
- tan(120°) = -√3
- csc(120°) = 2/√3
- sec(120°) = -2
- cot(120°) = -1/√3
Angle 135°
- sin(135°) = √2/2
- cos(135°) = -√2/2
- tan(135°) = -1
- csc(135°) = √2
- sec(135°) = -√2
- cot(135°) = -1
Angle 150°
- sin(150°) = 1/2
- cos(150°) = -√3/2
- tan(150°) = -1/√3
- csc(150°) = 2
- sec(150°) = -2/√3
- cot(150°) = -√3
Angle 180°
- sin(180°) = 0
- cos(180°) = -1
- tan(180°) = 0
- csc(180°) = Undefined
- sec(180°) = -1
- cot(180°) = Undefined
Angle 270°
- sin(270°) = -1
- cos(270°) = 0
- tan(270°) = Undefined
- csc(270°) = -1
- sec(270°) = Undefined
- cot(270°) = 0
These values can be derived using the unit circle or by utilizing trigonometric identities. Each angle corresponds to a specific position on the unit circle, allowing us to determine the sine (y-coordinate) and cosine (x-coordinate), from which the other ratios can be calculated.