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 θ and sec θ.'
A. csc θ = 5/4, sec θ = 4/5
B. csc θ = 5/4, sec θ = 5/3
C. csc θ = 4/5, sec θ = 5/3
D. csc θ = 4/5, sec θ = 3/5
We can use the Pythagorean theorem to find the length of the hypotenuse:
h = sqrt(x^2 + y^2) = sqrt(6^2 + 8^2) = 10
Then, we can use the definitions of the trigonometric functions:
sin θ = y/h = 8/10 = 4/5
cos θ = x/h = 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
Therefore, the answer is B. csc θ = 5/4, sec θ = 5/3.
To find the values of csc θ and sec θ for the point (6, 8), we can use the Pythagorean theorem.
First, let's calculate the hypotenuse of the right triangle formed by the point (6, 8). The hypotenuse can be found using the formula:
hypotenuse = sqrt(x^2 + y^2)
Substituting the values, we get:
hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10
Next, we can determine the value of csc θ using the formula:
csc θ = hypotenuse / y
Substituting the values, we get:
csc θ = 10 / 8 = 5/4
Finally, we can determine the value of sec θ using the formula:
sec θ = hypotenuse / x
Substituting the values, we get:
sec θ = 10 / 6 = 5/3
Therefore, the values of csc θ and sec θ for the point (6, 8) are:
csc θ = 5/4 and sec θ = 5/3
So, the correct answer is B. csc θ = 5/4, sec θ = 5/3.