(a) sin θ = 0.9263 θ = 67.9°, 112.1°
(b) cos θ = â’0.6909 θ = 133.7°, 226.3°
(c) tan θ = â’1.5416 θ = 123.0°, 303.0°
(d) cot θ = 1.3952 θ = 35.6°, 215.6°
(e) sec θ = 1.4293 θ = 45.6°, 225.6°
(f) csc θ = â’2.3174 θ = 205.6°, 334.4°
This is simply a matter of knowing how to use the trig identities and reflections. I am going to assume that you have access to an arctangent function (the ability to get the angle from the tangent of the angle) and that you have no other inverse trig functions available. The arctangent function is assumed to only work for positive tangents and returns a value between 0 and 90 degrees.
(a) sin θ = 0.9263 θ = 67.9°, 112.1°
The cos will be sqrt(1-0.9263^2) = 0.3768
The tan will be 0.9263/0.3768 = 2.4583
atan(2.4583) = 67.9°
Since the sin is positive, there are 2 angles, one in quadrant 1 and another in quadrant 2. The angle for quadrant 2 will be
180° - 67.9° = 112.1°
(b) cos θ = â’0.6909 θ = 133.7°, 226.3°
The cos is negative, but we'll use the positive value for the basic angle calculations.
sin = sqrt(1-0.6909^2) = 0.7230
tan = 0.7230/0.6909 = 1.0465
atan(1.0465) = 46.3°
Since the cos is negative, the angles are in quadrants II and III. The angles will be
180° - 46.3° = 133.7°
180° + 46.3° = 226.3°
(c) tan θ = â’1.5416 θ = 123.0°, 303.0°
atan(1.5416) = 57.0°
Since the tangent is negative, the angles are in quadrants II and IV.
180° - 57.0° = 123.0°
360° - 57.0° = 303.0°
(d) cot θ = 1.3952 θ = 35.6°, 215.6°
tan = 1/1.3952 = 0.7167
atan(0.7167) = 35.6°
Since the cot is positive, the angles are in quadrants I and III
180° + 35.6° = 215.6°
(e) sec θ = 1.4293 θ = 45.6°, 225.6°
cos = 1/1.4293 = 0.6996
sin = sqrt(1-0.6996^2) = 0.7145
tan = 0.7145/0.6996 = 1.0213
atan(1.0213) = 45.6°
Since the sec is positive, the angles are in quadrants I and III
180° + 45.6° = 225.6°
(f) csc θ = â’2.3174 θ = 205.6°, 334.4°
sin = 1/2.3174 = 0.4315
cos = sqrt(1-0.4315^2) = 0.9021
tan = 0.4315/0.9021 = 0.4783
atan(0.4783) = 25.6°
Since the csc is negative, the angles are in quadrants III and IV
180° + 25.6° = 205.6°
360° - 25.6° = 334.4°
