Questions: Select the correct choices that complete the sentence below. If the terminal side of an angle θ lies in quadrant III, then the values of tan θ and cot θ are and all other trigonometric function values are

Solution Steps

In trigonometry, the signs of trigonometric functions depend on the quadrant in which the terminal side of an angle lies. In quadrant III, both the x-coordinate and y-coordinate of a point on the terminal side are negative.

The tangent function, \(\tan \theta\), is defined as the ratio of the y-coordinate to the x-coordinate (\(\tan \theta = \frac{y}{x}\)). In quadrant III, both y and x are negative, so their ratio is positive. Therefore, \(\tan \theta\) is positive in quadrant III.

The cotangent function, \(\cot \theta\), is the reciprocal of the tangent function (\(\cot \theta = \frac{x}{y}\)). Since both x and y are negative, \(\cot \theta\) is also positive in quadrant III.

In quadrant III, the sine function (\(\sin \theta = \frac{y}{r}\)) and the cosine function (\(\cos \theta = \frac{x}{r}\)) are both negative because y and x are negative, while the hypotenuse \(r\) is always positive. Consequently, the secant (\(\sec \theta = \frac{1}{\cos \theta}\)) and cosecant (\(\csc \theta = \frac{1}{\sin \theta}\)) functions are also negative.

Final Answer