The ASTC or CAST or unit circle method for finding two angles having a given sine, cosine or tangent
In this method you draw a set of axes and label the quadrants 1, 2, 3 and 4
with the letters A, S, T and C respectively as shown in the diagrams below.
The letters mean:
- A: all three functions, sine, cosine and tangent are positive
in this quadrant
- S: only the sine function is positive in this quadrant
- T: only the tangent function is positive in this quadrant
- C: only the cosine function is positive in this quadrant
Next you draw two triangles in standard position
on the diagram. The two triangles are congruent (identical) and must be drawn
in the two quadrants that have the correct sign for your function.
The various diagrams below show all the possible combinations. Make sure that
you understand each of them. Then you use your calculator to get the
first angle, θPV , and finally you use the symmetry
of the diagram to get the second angle, θ2 .
CAST diagrams for the sine function
(see instructions)
CAST diagrams for the cosine function
(see instructions)
CAST diagrams for the tangent function
(see instructions)
Example: Find the two angles θ between 0° and 360°
for which cos(θ) = −0.4.
- The CAST diagram for negative cosine is as shown.
- In degree mode the calculator or Algebra Coach gives
θPV = arccos(−0.4) = 113.6°
- The symmetry of the diagram gives
θ2 = 360° − 113.6°
= 246.4°
Thus the answers are 113.6° and 246.4°.
Example: Find the two angles θ between 0 and 2π
radians for which tan(θ) = −3.8.
- The CAST diagram for negative tangent is as shown.
- The calculator or Algebra Coach gives
θPV = arctan(−3.8) =
−1.313 radians
- The symmetry of the diagram gives
θ2 = −1.313 + π
= 1.828 radians
Because we want answers between 0 and 2π we ‘correct’
θPV by adding 2π (to get 4.97)
and state that the answers are 1.828 radians and 4.97 radians
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