Problem 29 - Algebra 2

Question

For each angle \(\theta,\) find the values of \(\cos \theta\) and \(\sin \theta .\) Round your answers to the nearest hundredth. $$ 0^{\circ} $$

Step-by-Step Solution

Answer

\(\cos 0^{\circ} = 1.00 , \sin 0^{\circ} = 0.00\)

Step 1: Find the value of the cosine function

Recalling from trigonometric definitions, \(\cos 0^{\circ} = 1\). Therefore the cosine of a 0 degree angle is 1.

Step 2: Find the value of the sine function

Again, remembering trigonometric rules, \(\sin 0^{\circ} = 0\). Hence, the sine of a 0 degree angle is 0.

Step 3: Round the values to the nearest hundredth

The given values are already at the maximum degree of precision required. Therefore, Cosine = 1.00 and Sine = 0.00 when rounded to the nearest hundredth.

Key Concepts

Cosine

The cosine function is an essential part of trigonometry. It represents the x-coordinate of a point on a unit circle or the adjacent side over the hypotenuse in a right triangle.
The function is often denoted as \( ext{cos}\), and it's evaluated using angles, typically measured in degrees or radians. This function varies between -1 and 1 for all possible inputs.

• A key characteristic of the cosine is its periodic nature; it repeats every 360° (or \(2\pi\) radians).
• At critical points: at \(0°\) (and each 360° increment), \( ext{cos}\) equals 1.
• At \(90°\) and \(270°\), it's 0, and at \(180°\), \( ext{cos}\) becomes -1.

These values and periodicity are crucial for solving trigonometric functions in various applications, including physics and engineering. Understanding them allows easy computation of the cosine function for any given angle.

Sine

The sine function is another fundamental trigonometric function, representing the y-coordinate of a point on the unit circle or the opposite side over the hypotenuse in a right triangle.
Denoted as \( ext{sin}\), the sine function, similar to cosine, oscillates between specific ranges, from -1 to 1.

• Sine also has a notable periodicity; it completes a full cycle every 360° (or \(2\pi\) radians).
• For reference, at \(0°\) and \(180°\) (and other multiple increments), \( ext{sin}\) equals 0.
• Conversely, at \(90°\), \( ext{sin}\) rises to 1, and at \(270°\), it dips to -1.

Understanding these critical points aids learners in visualizing how angles translate to specific sine values, crucial for calculating wave functions, pendulum movements, and alternating currents in various fields of study.

Angles

Angles are fundamental to understanding trigonometry and geometry. They are measures of rotation and can be quantified in degrees or radians, depending on context.
Angles serve as the base input for trigonometric functions such as sine and cosine.

• Common angles such as \(0°, 30°, 45°, 60°,\) and \(90°\) are frequently used in trigonometry due to their simple trigonometric values.
• Trigonometry often employs both degrees and radians; \(0°\) is equal to \(0\) radians, \(90°\) equals \(\frac{\pi}{2}\) radians, and a full circle of \(360°\) equals \(2\pi\) radians.

Understanding the relationship between angles and their trigonometric values is fundamental to solving geometric problems, whether on a plane with simple vectors or in three dimensions with complex shapes.