Determine the values of the trigonometric functions of t if P(t) lies in the second quadrant and on the line
y = −2x.
To determine the values of the trigonometric functions of t if P(t) lies on the line y = -2x and in the second quadrant, we need to find the angle t in the second quadrant that corresponds to the point P(t).
First, let's rewrite the equation y = -2x in terms of trigonometric functions. We know that tan(t) = y/x, so we can write -2 = tan(t).
To find the angle t in the second quadrant, we need to find the inverse tangent of -2. We can use a calculator or table of values to find that the angle t is approximately -63.43 degrees or -1.11 radians. Remember that angles in the second quadrant have values between 90 and 180 degrees or π/2 and π radians.
Now, we can determine the values of the trigonometric functions of t:
- sin(t) = sin(-63.43°) or sin(-1.11) ≈ -0.8944
- cos(t) = cos(-63.43°) or cos(-1.11) ≈ 0.4470
- tan(t) = -2 (from the equation y = -2x)
- csc(t) = 1/sin(t) ≈ -1.1180
- sec(t) = 1/cos(t) ≈ 2.2361
- cot(t) = 1/tan(t) ≈ -0.5
So, the values of the trigonometric functions of t if P(t) lies on the line y = -2x in the second quadrant are approximately:
sin(t) ≈ -0.8944
cos(t) ≈ 0.4470
tan(t) = -2
csc(t) ≈ -1.1180
sec(t) ≈ 2.2361
cot(t) ≈ -0.5