Questions: What are the exact values of sin θ, cos θ, and tan θ for an angle θ in standard position terminating at point P(2,-12)?

Solution Steps

To find the trigonometric values for an angle \(\theta\) with a terminal point \(P(2, -12)\), we first calculate the distance from the origin to the point, which is the hypotenuse of the right triangle formed. This is done using the Pythagorean theorem. Then, we use the definitions of sine, cosine, and tangent in terms of the coordinates and the hypotenuse.

To find the hypotenuse \( r \) of the right triangle formed by the point \( P(2, -12) \), we use the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} = \sqrt{2^2 + (-12)^2} = \sqrt{4 + 144} = \sqrt{148} \approx 12.1655 \]

The sine of the angle \( \theta \) is given by: \[ \sin \theta = \frac{y}{r} = \frac{-12}{12.1655} \approx -0.9864 \]

The cosine of the angle \( \theta \) is given by: \[ \cos \theta = \frac{x}{r} = \frac{2}{12.1655} \approx 0.1644 \]

The tangent of the angle \( \theta \) is given by: \[ \tan \theta = \frac{y}{x} = \frac{-12}{2} = -6.0 \]

Final Answer