Question

Approximate, to the nearest 0.1°, all angles θ in the interval [0°, 360°) that satisfy the equation. (Enter your answers as a comma-separated list.)

(a) sin θ = 0.9263 θ = °
(b) cos θ = −0.6909 θ = °
(c) tan θ = −1.5416 θ = °
(d) cot θ = 1.3952 θ = °
(e) sec θ = 1.4293 θ = °
(f) csc θ = −2.3174

Answer 1

(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°