Problem 4 - Precalculus

Question

Find the exact value of the cosine and sine of the given angle. $$ \theta=\frac{\pi}{2} $$

Step-by-Step Solution

Answer

\(\cos(\frac{\pi}{2}) = 0, \sin(\frac{\pi}{2}) = 1\)

Step 1: Identify the Angle Position on the Unit Circle

We need to locate where \(\theta = \frac{\pi}{2}\) is positioned on the unit circle. The angle \(\frac{\pi}{2}\) radians is equivalent to 90 degrees, which places it at the positive y-axis, the topmost point on the unit circle.

Step 2: Determine the Coordinates on the Unit Circle

The unit circle has a radius of 1. At \(\frac{\pi}{2}\) radians, the coordinates are \((0, 1)\). This represents a point directly upward along the y-axis.

Step 3: Identify Sine and Cosine Values from Coordinates

On the unit circle, the x-coordinate gives the cosine value, and the y-coordinate gives the sine value. Here, the coordinates are \((0, 1)\).

Step 4: Extract Cosine of the Angle

From the coordinates \((0, 1)\), the cosine of \(\theta = \frac{\pi}{2}\) is the x-coordinate, which is 0.

Step 5: Extract Sine of the Angle

From the coordinates \((0, 1)\), the sine of \(\theta = \frac{\pi}{2}\) is the y-coordinate, which is 1.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 centered at the origin of a coordinate plane. Simply put, it helps us understand how the trigonometric functions work. When you imagine a circle on a graph,
the unit circle allows us to represent angles and their corresponding coordinates easily.

• Every point on this circle is described by \(x\) and \(y\) coordinates,
  which are determined by the trigonometric functions cosine and sine, respectively.
• The unit circle essentially connects geometric concepts with trigonometric functions,
  helping to visualize angles and their related values.

The beauty of the unit circle is that it allows for an easy and systematic way
to determine the trigonometric values of common angles.

Cosine Function

The cosine function is one of the primary trigonometric functions, represented as \( \cos(\theta) \). In the context of the unit circle, it describes the x-coordinate of a point at a given angle \(\theta\). Understanding this function is crucial
when working with angles in trigonometry.

• For angle \( \theta \), \( \cos(\theta) \) tells us how "wide" the angle opens from the x-axis,
  providing a horizontal measurement of its impact on the circle.
• In a right-angled triangle, \( \cos(\theta) \) represents the ratio of the
  adjacent side to the hypotenuse.

In our exercise, \( \cos(\theta = \frac{\pi}{2}) = 0 \), indicating the angle is perfectly perpendicular
to the x-axis in the coordinate plane.

Sine Function

The sine function, symbolized as \( \sin(\theta) \), is a foundational trigonometric function that describes
the y-coordinate of a point at an angle \( \theta \) on the unit circle.
It plays a significant role in determining the vertical orientation of angles.

• In essence, \( \sin(\theta) \) describes how tall or short the angle extends from the origin of the circle.
• Within a right-angled triangle, the sine function is the ratio of the opposite side to the hypotenuse.

For \( \theta = \frac{\pi}{2} \, \sin(\theta) = 1 \),
indicating the maximum upward stretch on the unit circle's y-axis.

Radian Measure

Radian measure is a way of expressing angles based on the radius of a circle. It provides a natural approach to measure angles using the idea of the arc length. When you think of radians,
remember they offer a different perspective than degrees, but represent the same concept of rotation.

• In mathematics, 1 radian is approximately 57.3 degrees.
  A full circle encompasses \( 2\pi \) radians, or 360 degrees.
• Radians allow for more straightforward calculations in calculus
  and other advanced branches of mathematics due to their relation to arc length.

For the unit circle, \( \frac{\pi}{2} \) radians is a quarter-turn, perfectly placing the angle along the positive y-axis.