Answer:
To find the values of the trigonometric functions for an angle θ, given that cos(θ) = 4/15 and sin(θ) < 0, we can use the Pythagorean identity and the information about the signs of sine and cosine in different quadrants.
1. Start by finding the value of sin(θ):
Since sin(θ) < 0, we know that θ is in either the third or fourth quadrant of the unit circle. In these quadrants, sine is negative.
Using the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1
Plugging in the value of cos(θ) = 4/15:
sin^2(θ) + (4/15)^2 = 1
sin^2(θ) + 16/225 = 1
sin^2(θ) = 1 - 16/225
sin^2(θ) = 209/225
Taking the square root of both sides: sin(θ) = -√(209/225)
Simplifying further: sin(θ) = -√209/15
2. Next, we can find the values of the other trigonometric functions:
tan(θ) = sin(θ)/cos(θ)
tan(θ) = (-√209/15)/(4/15)
Simplifying further: tan(θ) = -√209/4
csc(θ) = 1/sin(θ)
csc(θ) = 1/(-√209/15)
Simplifying further: csc(θ) = -15/√209
sec(θ) = 1/cos(θ)
sec(θ) = 1/(4/15)
Simplifying further: sec(θ) = 15/4
cot(θ) = 1/tan(θ)
cot(θ) = 1/(-√209/4)
Simplifying further: cot(θ) = -4/√209
So, the values of the trigonometric functions are as follows:
sin(θ) = -√209/15
cos(θ) = 4/15
tan(θ) = -√209/4
csc(θ) = -15/√209
sec(θ) = 15/4
cot(θ) = -4/√209
Explanation: