Problem 23 - Algebra 2 and Trigonometry

Question

What is the smallest positive value of \(\theta\) such that \(\cos \theta=0 ?\)

Step-by-Step Solution

Answer

The smallest positive value of \(\theta\) is \(\pi/2\).

Step 1: Understanding the Problem

We are asked to find the smallest positive value of \(\theta\) for which \(\cos \theta = 0\). This means we are looking for an angle between \(0\) and \(360\) degrees (\(0\) to \(2\pi\) radians) where the cosine of the angle is zero.

Step 2: Recalling Trigonometric Facts

The function \(\cos \theta\) represents the cosine of an angle \(\theta\). Cosine equals zero at specific points on the unit circle, which correspond to the \(\theta\) values where the x-coordinate on the unit circle is zero.

Step 3: Identifying the Correct Quadrants

The cosine function equals zero when the angle \(\theta\) is an odd multiple of \(\pi/2\) (90 degrees). This occurs because at these points, the terminal side of the angle is aligned with the y-axis on the unit circle.

Step 4: Finding the Smallest Positive Solution

The smallest positive angle for which \(\cos \theta = 0\) is at \(\theta = \pi/2\), also known as 90 degrees.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \theta \), is a fundamental concept in trigonometry. It is linked to the x-coordinate of a point on the unit circle corresponding to an angle \( \theta \). The cosine value indicates how far along the x-axis (horizontal axis) a point is.

• When \( \cos \theta = 1 \), the angle \( \theta \) is such that the point is all the way to the right on the circle, making the x-coordinate 1 — this occurs at 0 or \( 2\pi \) radians.
• When \( \cos \theta = -1 \), the point is all the way to the left, with the x-coordinate -1, typically at \( \pi \) radians.
• When \( \cos \theta = 0 \), the point sits directly on the y-axis, neither to the left nor right of the origin. This scenario is found at odd multiples of \( \frac{\pi}{2} \).

Thus, the cosine function can help us determine directions and positions of an angle's terminal side on the unit circle. This function is periodic, repeating its values in a predictable pattern every \( 2\pi \) radians.

Unit Circle

The unit circle is one of the most powerful tools in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. The simplicity of its radius being 1 makes it easy to relate trigonometric functions to angles and their respective coordinates.

• The circle's equation is \( x^2 + y^2 = 1 \), where \( (x, y) \) are the coordinates of a point on the circle.
• An angle \( \theta \) measured in radians starts from the positive x-axis and moves counter-clockwise around the circle. This movement helps visualize trigonometric relationships.
• For each angle \( \theta \), \( \cos \theta \) corresponds to the x-coordinate and \( \sin \theta \) corresponds to the y-coordinate of the point on the circle.

The unit circle helps solve problems by providing a graphical representation of trigonometric functions, highlighting how angles relate to coordinates. If \( \cos \theta = 0 \), the corresponding angles are where the circle intersects the y-axis: odd multiples of \( \frac{\pi}{2} \).

Angles in Radians

Angles can be measured in different units: degrees and radians. While degrees are intuitive, radians are essential in mathematics, especially trigonometry.

• Radians are a measure of angle based on the radius of a circle. An angle of 1 radian means the arc length equals the radius.
• There are \( 2\pi \) radians in a full circle, which corresponds to 360 degrees. Therefore, \( \pi \) radians is equivalent to 180 degrees.
• Radians provide a natural way to describe angles in mathematical equations and functions, as they correspond directly to the geometry of a circle.

Understanding radians becomes crucial when solving trigonometric equations like \( \cos \theta = 0 \). Here, the solution of \( \theta = \frac{\pi}{2} \) radians is the smallest positive angle at which this condition is satisfied.