Problem 8 - Precalculus : Building Concepts and Connections

Question

Find exact values of the given trigonometric functions without the use of a calculator. $$\arcsin 0$$

Step-by-Step Solution

Answer

The exact value of \(\arcsin(0)\) is 0 degrees

Step 1: Understand the unit circle and the sine function

The unit circle is a circle with a radius of 1 and its center at the origin (0,0) in the Cartesian coordinate system. The sine function gives the height (or the y-coordinate) of a point on the unit circle. It depicts the ratio of the length of the side opposite a given angle in a right triangle to the length of the hypotenuse. It can take on any value from -1 to 1. For angle \(\theta\), \(\sin \theta = 0\) at \(\theta = 0\) or \(\theta = 180\) degrees. Since the range of \(\arcsin\) is -90 to 90 degrees, the \(\arcsin(0)\) will return 0 degrees.

Step 2: Apply arcsin

Knowing that sine of 0 degrees is 0, when we reverse this process we get that \(\arcsin(0)\) equals 0 degrees.

Key Concepts

Unit Circle

Understanding the unit circle is foundational for mastering trigonometry. Imagine a perfect circle with its center positioned at (0,0) in the coordinate system and a radius of 1. This is the unit circle. It acts as a reference for measuring angles and defining sine, cosine, and other trigonometric functions.

To visualize how the unit circle works, think about drawing various triangles by intersecting radius lines with the circumference. The coordinates of any point on the circle are corresponding to the cosine and sine of the angle formed by the radius in which it lies. Thus, they range from -1 to 1.

• The x-coordinate at any point gives the cosine value of the angle.
• The y-coordinate provides the sine value.

Since the unit circle is centered at the origin, every point represents the hypotenuse of a right triangle where the radius is 1.

Sine Function

The sine function is vital in trigonometry and cyclic behavior applications, like waves. It captures how the y-coordinate of a point on the unit circle shifts as the angle changes.

In essence, the sine of an angle θ is the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. While this initially pertains to triangles, the unit circle allows us to extend this idea to all real numbers by following the circle's circumference. For angles within a full rotation (0 to 360 degrees), the sine function will have values between -1 and 1.

An essential property is that the sine function fluctuates smoothly and periodically. The prime examples to remember are:

• \(\sin(0) = 0\)
• \(\sin(90^\circ) = 1\)
• \(\sin(180^\circ) = 0\)
• \(\sin(270^\circ) = -1\)

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of a triangle to the ratios of its sides. The primary trigonometric functions are sine, cosine, and tangent.

• Sine (sin): Gives the ratio of the opposite side to the hypotenuse.
• Cosine (cos): Provides the ratio of the adjacent side to the hypotenuse.
• Tangent (tan): Represents the ratio of the opposite side to the adjacent side.

These functions are cyclic, which means they repeat their values in predictable intervals. This quality is useful in physically modeling waves, oscillations, and circular motion.

Knowing each function's range and periodic behavior helps to predict outcomes and solve equations, such as finding specific angle values with \(\arcsin\) and other inverse functions. This is crucial in fields as varied as engineering, physics, and even music.