Questions: A point on the terminal side of angle θ is given. Find the exact value of the indicated trigonometric function of θ. (-3,-2) Find sec θ

Solution Steps

To find \(\sec \theta\), we first need to determine the hypotenuse of the right triangle formed by the point \((-3, -2)\) and the origin. The hypotenuse is the distance from the origin to the point, which can be found using the Pythagorean theorem. Once we have the hypotenuse, we can find \(\cos \theta\) as the ratio of the x-coordinate to the hypotenuse. Finally, \(\sec \theta\) is the reciprocal of \(\cos \theta\).

To find the hypotenuse of the right triangle formed by the point \((-3, -2)\) and the origin, we use the Pythagorean theorem: \[ \text{hypotenuse} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \] The calculated hypotenuse is approximately \(3.6056\).

The cosine of the angle \(\theta\) is given by the ratio of the x-coordinate to the hypotenuse: \[ \cos \theta = \frac{-3}{\sqrt{13}} \approx -0.8321 \]

The secant of the angle \(\theta\) is the reciprocal of the cosine: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-0.8321} \approx -1.2019 \]

Final Answer