Question

If \(\theta\) is an angle in standard position, state in what quadrants its terminal side can lie if \(\sec \theta\) is positive.

Short answer

\(\sec \theta\) is positive in the 1st and 4th quadrants.

Step-by-step solution

Understand Secant Function

Recall that \(\sec \theta\) is the reciprocal of the cosine function. Therefore, it is positive where the cosine function is positive.

Identify the Quadrants Where Cosine is Positive

The cosine of an angle is positive in the 1st and 4th quadrants because it corresponds to the x-coordinate on the unit circle, which is positive in these quadrants.

Conclude the Quadrants for Positive Secant

Since the secant is the reciprocal of cosine and takes the same sign, \(\sec \theta\) is positive in the same quadrants where cosine is positive – the 1st and 4th quadrants.

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in the study of geometry, particularly in the context of angles and circles. Among these functions, we find sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent.

Each of these functions relates an angle to a ratio of sides in a right-angled triangle or to a coordinate on the unit circle. In the case of the secant function, which is of particular interest here, it is defined as the reciprocal of the cosine function, or \( \sec \theta = \frac{1}{\cos \theta} \) when \(\cos \thetaeq 0\).

The sign of the trigonometric functions, including secant, depend on the quadrant in which the terminal side of an angle in standard position lies. For example, in the first quadrant, where all angles are between 0 and 90 degrees, all sine, cosine, and tangent values are positive. Understanding these positivity/negativity rules is essential for solving trigonometric equations and for graphing these functions.

Unit Circle

The unit circle is a powerful tool for understanding and visualizing the behavior of trigonometric functions. It's a circle centered at the origin of a coordinate plane with a radius of 1 unit.

Each point on the unit circle has coordinates that correspond to the cosine and sine values of an angle whose terminal side passes through that point. Therefore, the coordinates are in the form of \( (\cos \theta, \sin \theta) \). The x-coordinate represents the cosine of the angle, and hence, is directly related to the secant function, as secant is the reciprocal of cosine.

As mentioned in the exercises, the secant function is positive when its corresponding cosine function is positive. That happens in the first quadrant, where both coordinates are positive, and the fourth quadrant, where the x-coordinate is positive, and the y-coordinate (sine function) is negative.

Standard Position of an Angle

Angles in trigonometry are often placed in a 'standard position' to simplify understanding and calculations. An angle is in standard position when its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis.

The angle's terminal side rotates counterclockwise from the initial side for positive angles, or clockwise for negative angles. This terminal side can lie in any of the four quadrants, determining the signs of the trigonometric function values for the given angle.

In the context of our problem, knowing the angle is in standard position helps us identify which quadrants to consider when determining the sign of the secant function. Since secant is positive in the same quadrants as the cosine function, and the cosine function is positive in the 1st and 4th quadrants, it follows that for an angle in standard position, the terminal side can only lie in those quadrants if the secant of the angle is positive.