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: 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. 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. 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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