Each point P is on the terminal side of an angle θ.
1. P(2,9) 2. P(5, -2)
a. Draw a diagram showing θ in standard position.
b. Calculate sin θ, cos θ, and tan θ in exact values.
c. How far is P from the origin?
d. What is the value of θ?
e. Give 2 positive and 1 negative co-terminal values of θ in standard position.
for (2,9)
r = √(2^2 + 9^2) = √85
sinθ = y/r = 9/√85
cosθ = x/r
and so on
θ = sin^-1(9/√85) = 77.47°
co-terminals are just θ±360k° for integer values of k.
for a standard position angle determined by the point (x,y) what are the values of the trigonometric functions for the point (6,8) find csc theta sec theta
First, we need to find r, which is the distance between the origin (0,0) and (6,8):
r = √(6^2 + 8^2) = √100 = 10
Next, we can find sin, cos, tan, csc, sec, and cot:
sin θ = y/r = 8/10 = 4/5
cos θ = x/r = 6/10 = 3/5
tan θ = y/x = 8/6 = 4/3
csc θ = 1/sin θ = 5/4
sec θ = 1/cos θ = 5/3
cot θ = 1/tan θ = 3/4
Finally, we can find csc θ × sec θ:
csc θ × sec θ = (1/sin θ) × (1/cos θ) = 5/4 × 5/3 = 25/12
Therefore, csc θ × sec θ = 25/12.
for a standard position angle determined by the point (x,y) what are the values of the trigonometric functions for the point (6,8) find csc theta sec theta
csc theta=5/4, sec theta=4/5
csc theta=5/4, sec theta=5/3
csc theta=4/5, sec theta 5/3
csc theta=4/5, sec theta 3/5
The correct answer is:
csc theta = 5/4, sec theta = 3/5.
for a standard position angle determined by the point (x,y) what are the values of the trigonometric functions for the point (6,8) find csc theta sec theta EXPLAIN
A.csc theta=5/4, sec theta=4/5
B.csc theta=5/4, sec theta=5/3
C.csc theta=4/5, sec theta =5/3
D.csc theta=4/5, sec theta= 3/5
To find the correct answer, we first need to determine the reference angle for the angle determined by the point (6,8) in standard position:
r = sqrt(6^2 + 8^2) = 10
sin(theta) = 8/10 = 4/5
cos(theta) = 6/10 = 3/5
The point (6,8) is in the second quadrant, so the reference angle will be:
theta_ref = pi/2 - sin^-1(4/5) = pi/2 - 0.93 = 0.64
Therefore, the values of the trigonometric functions for an angle which contains the point (6,8) in standard position are:
sin(theta) = sin(pi - theta) = -sin(0.64) ≈ -0.60
cos(theta) = cos(pi - theta) = -cos(0.64) ≈ -0.79
tan(theta) = tan(pi - theta) = sin(0.64) / -cos(0.64) ≈ 0.76 / -0.65 ≈ -1.17
csc(theta) = 1 / sin(theta) ≈ -1.67 / -0.60 ≈ 2.78
sec(theta) = 1 / cos(theta) ≈ -1.26 / -0.79 ≈ 1.59
cot(theta) = 1 / tan(theta) ≈ -0.87 / 0.76 ≈ -1.15
Therefore, the correct answer is option D. csc theta = 4/5, sec theta = 3/5.