Given Cos= -5/13 where 0<=theta<=360.
a) In which quadrant(s) would the terminal arm of theta lie?
b)Determine all possible primary trig ratios for theta?
c) Evaluate all possible values of theta to the nearest degree.
a) The cosine function is negative in the second and third quadrants. Therefore, the terminal arm of theta would lie in either the second or the third quadrant.
b) Since the cosine function is equal to -5/13, we can determine the values of the other primary trigonometric ratios using the Pythagorean identity. The Pythagorean identity states that cos^2(theta) + sin^2(theta) = 1.
Solving for sin^2(theta), we have:
sin^2(theta) = 1 - cos^2(theta)
sin^2(theta) = 1 - (-5/13)^2
sin^2(theta) = 1 - 25/169
sin^2(theta) = 144/169
sin(theta) = ±√(144/169)
sin(theta) = ±12/13
Using the values of cos(theta) and sin(theta), we can determine the values of the other primary trigonometric ratios:
tan(theta) = sin(theta)/cos(theta) = (±12/13)/(-5/13) = ±12/5
csc(theta) = 1/sin(theta) = 1/(±12/13) = ±13/12
sec(theta) = 1/cos(theta) = 1/(-5/13) = -13/5
cot(theta) = 1/tan(theta) = 1/(±12/5) = ±5/12
Therefore, the possible primary trig ratios for theta are:
sin(theta) = ±12/13
cos(theta) = -5/13
tan(theta) = ±12/5
csc(theta) = ±13/12
sec(theta) = -13/5
cot(theta) = ±5/12
c) To find the possible values of theta, we can use the inverse cosine function. Taking the inverse cosine of -5/13, we have:
θ = cos^(-1)(-5/13)
θ ≈ 116.57 degrees
Since the cosine function is negative in both the second and third quadrants, there is another possible value for theta in the third quadrant. To find this value, we can subtract the previous angle from 360 degrees:
θ = 360 - 116.57
θ ≈ 243.43 degrees
Therefore, the possible values of theta to the nearest degree are approximately 117 degrees and 243 degrees.