Problem 351 - Precalculus

Question

For the following exercises, use side lengths to evaluate. $$ \cos \left(\frac{\pi}{2}\right)=\sin \left(\\_^{\circ}\right) $$

Step-by-Step Solution

Answer

\( \sin(0^{\circ}) = 0 \).

Step 1: Understand the cosine value

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. The angle \( \frac{\pi}{2} \) is equivalent to 90 degrees. Therefore, \( \cos \left( \frac{\pi}{2} \right) = \cos(90^{\circ}) = 0 \). This is because at 90 degrees, there is no adjacent side compared to the hypotenuse.

Step 2: Determine the equivalent sine value

The sine of an angle is the ratio of the opposite side to the hypotenuse. If we need to match the value of \( \cos \left( \frac{\pi}{2} \right) = 0 \) to a sine value, we need to find an angle whose sine is also 0.

Step 3: Identify the sine angle

The sine of 0 degrees or 180 degrees is 0, since at both these angles, the opposite side is 0. Therefore, \( \sin(\alpha) = 0 \) at \( \alpha = 0^{\circ} \) or \( \alpha = 180^{\circ} \). However, typically, the principal angle equivalent to \( \cos(90^{\circ}) \) is \( \sin(0^{\circ}) \), so \( \_^{\circ} = 0^{\circ} \).

Key Concepts

Understanding Cosine

The cosine function is a fundamental concept in trigonometry, especially when dealing with right triangles. It's all about the ratio of the side lengths. In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. This can be written as:

• Cosine: \( rac{\text{Adjacent side}}{\text{Hypotenuse}} \)

When considering the angle \( \frac{\pi}{2} \) or 90 degrees, something special happens. The adjacent side becomes nonexistent or zero in terms of length, relative to the hypotenuse.
This results in \( \cos(90^{\circ}) = 0 \). At 90 degrees, you are essentially standing straight up from the adjacent side, making its influence on the cosine value null. Understanding this key point helps build the foundation for more complex trigonometric concepts.

Understanding Sine

The sine function works closely with the cosine function but focuses on a different aspect of the right triangle. The sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Therefore, it can be expressed as:

• Sine: \( \frac{\text{Opposite side}}{\text{Hypotenuse}} \)

For a right triangle, when you consider an angle of 0 degrees, the opposite side to the angle virtually disappears. As a result, the sine value at 0 or 180 degrees is zero.
This is written as \( \sin(0^{\circ}) = 0 \). Sine reflects how "vertically" an angle pushes away from the horizontal baseline. At 0 or 180 degrees, there's no opposite push, making its value zero.

Right Triangle Ratios

In trigonometry, understanding the relationships between the sides of a right triangle is crucial. The ratios—sine, cosine, and tangent—help us figure out these relationships using angles. They provide a toolkit to solve for unknown sides and angles in diverse scenarios.
Two critical ratios, as earlier discussed, include:

• Cosine: Adjacent side to hypotenuse
• Sine: Opposite side to hypotenuse

These ratios allow us to take raw side measurements and convert them into insight about angles, and vice versa. When angles are 90 degrees or zero, these ratios reveal unique boundary conditions, indicating parallel alignment with a particular side.
By mastering right triangle ratios, students unlock the ability to navigate through trigonometric identities and equations, applying these principles in calculus, physics, and beyond. The interplay between angles and side lengths opens up a comprehensive view of geometric relationships.