Final answer:
The trigonometric function values for an angle whose terminal side passes through (6, -4) should be derived using the coordinates and the Pythagorean theorem. Since none of the provided options match these calculated values, all options A, B, C, and D are incorrect.
Step-by-step explanation:
The student's question involves determining the trigonometric function values for an angle θ whose terminal side passes through the point (6, -4).
In this context, the answers represent the sine (sin), cosine (cos), and tangent (tan) of the angle. To find these values, one should note that for an angle in standard position:
- The cosine (cos) of the angle is the x-coordinate divided by the hypotenuse.
- The sine (sin) of the angle is the y-coordinate divided by the hypotenuse.
- The tangent (tan) of the angle is the y-coordinate divided by the x-coordinate.
The hypotenuse (h) can be found using the Pythagorean theorem: √(x² + y²).
For the point (6, -4), the hypotenuse is √(6² + (-4)²) = √(36 + 16) = √52. Therefore, the correct trigonometric function values are:
- sin(θ) = -4/√52
- cos(θ) = 6/√52
- tan(θ) = -4/6 = -2/3
After simplifying, we get:
- sin(θ) = -4/(√52)
- cos(θ) = 6/(√52)
- tan(θ) = -2/3
These values should be rationalized by multiplying the numerator and denominator by √52 if necessary. Given the options provided by the student, none of them correctly represent the sin, cos, and tan values for the point (6, -4) based on the hypotenuse calculation. Therefore, all options A, B, C, and D are incorrect.