To find the values of the six trigonometric functions of theta (θ) where θ is in the second quadrant and sin(θ) = 35, we first note that the value of sin(θ) seems incorrect because the sine function can only take values between -1 and 1. However, if we consider that the problem might be intended to express sin(θ) in terms of a hypothetical triangle where the opposite side is 35 and the hypotenuse is 1 (a unique case), we can proceed with the calculations accordingly. We can use the Pythagorean theorem to find the adjacent side.
Let’s denote:
- Opposite side = 35
- Hypotenuse = 1
We use the Pythagorean theorem to find the adjacent side, represented as adj:
adj^2 + 35^2 = 1^2
This calculation seems invalid since we cannot have a valid triangle with these dimensions. Usually, we want sin(θ) to be a valid value between -1 and 1. If we were to use a reasonable trigonometric value, say sin(θ) = 0.35 instead, we can proceed with that to find the other functions in quadrant II.
Assuming sin(θ) = 0.35:
- sin(θ) = 0.35
- Since θ is in the second quadrant, cos(θ) will be negative and can be found using:
cos(θ) = -√(1 – sin²(θ))
Calculating this gives:
- sin²(θ) = (0.35)² = 0.1225
- cos(θ) = -√(1 – 0.1225) = -√(0.8775) ≈ -0.936
Now, we’ll calculate the other trigonometric functions:
- tan(θ) = sin(θ) / cos(θ) = 0.35 / -0.936 ≈ -0.374
- csc(θ) = 1/sin(θ) = 1/0.35 ≈ 2.857
- sec(θ) = 1/cos(θ) = 1/-0.936 ≈ -1.070
- cot(θ) = 1/tan(θ) = 1/-0.374 ≈ -2.673
So, the values of the six trigonometric functions for sin(θ) = 0.35 in quadrant II are approximately:
- sin(θ) ≈ 0.35
- cos(θ) ≈ -0.936
- tan(θ) ≈ -0.374
- csc(θ) ≈ 2.857
- sec(θ) ≈ -1.070
- cot(θ) ≈ -2.673
This completes the breakdown of finding the values of the six trigonometric functions of θ based on the given constraints.