Calculate Sin²(100°) + Cos²(80°). Calculate The Probability Of Choosing A Rational Number From The Set {∛1, ∛2, ∛3, ..., ∛27}.
In this comprehensive guide, we will delve into two intriguing mathematical problems. First, we will determine the value of sin²(100°) + cos²(80°). Then, we will calculate the probability of selecting a specific type of number from a given set. Let's embark on this mathematical journey together!
Determining the Value of sin²(100°) + cos²(80°)
To determine the value of sin²(100°) + cos²(80°), we will leverage trigonometric identities and transformations. Our primary goal is to simplify the expression and express it in terms of known trigonometric values. This problem beautifully showcases the power of trigonometric manipulation and the elegance of mathematical relationships.
Utilizing Trigonometric Identities
We can start by employing the trigonometric identity cos(90° - x) = sin(x). This identity allows us to relate the cosine of an angle to the sine of its complementary angle. Applying this identity to cos(80°), we get:
cos(80°) = cos(90° - 10°) = sin(10°)
Now, our expression becomes:
sin²(100°) + cos²(80°) = sin²(100°) + sin²(10°)
Next, we can use the identity sin(180° - x) = sin(x). This identity helps us to express the sine of an angle greater than 90° in terms of the sine of an acute angle. Applying this to sin(100°), we have:
sin(100°) = sin(180° - 80°) = sin(80°)
However, a more direct approach using sin(100°) = sin(180° - 100°) = sin(80°) leads to:
sin²(100°) = sin²(80°)
Substituting this back into our expression, we get:
sin²(100°) + sin²(10°) = sin²(80°) + sin²(10°)
Further Simplification and Calculation
To further simplify, we recall the identity sin(90° - x) = cos(x). Thus, sin(80°) = sin(90° - 10°) = cos(10°). Now our expression looks like this:
sin²(80°) + sin²(10°) = cos²(10°) + sin²(10°)
Here, we arrive at the fundamental trigonometric identity:
sin²(x) + cos²(x) = 1
Therefore, cos²(10°) + sin²(10°) = 1.
Conclusion for Trigonometric Calculation
In conclusion, by strategically applying trigonometric identities and transformations, we have successfully determined that:
sin²(100°) + cos²(80°) = 1
This result highlights the elegance and interconnectedness of trigonometric relationships. The use of identities allowed us to simplify a seemingly complex expression into a concise and meaningful value. Understanding and applying these principles is crucial in various fields of mathematics, physics, and engineering. Further practice and exploration will solidify your grasp of these concepts, enabling you to tackle a wide array of trigonometric problems with confidence and precision. Remember, the key to mastering trigonometry lies in recognizing patterns, applying identities, and persistently simplifying expressions until you reach a solution. This problem serves as an excellent illustration of how these techniques can be effectively employed to unravel mathematical complexities.
Calculating Probabilities from a Set of Radicals
Now, let's shift our focus to the second part of the problem: calculating the probability of selecting a specific type of number from a given set. This involves understanding the nature of the set, identifying favorable outcomes, and applying the basic principles of probability.
Defining the Set and the Problem
We are given the set {∛1, ∛2, ∛3, ..., ∛27} and we want to calculate the probability of selecting a rational number from this set. To do this, we must first understand what constitutes a rational number within this context. This problem provides a great opportunity to apply the concepts of number theory and probability in a practical scenario.
The set consists of the cube roots of integers from 1 to 27. A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. In the context of cube roots, a rational number will result when the number under the cube root is a perfect cube.
Identifying Favorable Outcomes
To find the favorable outcomes, we need to identify the perfect cubes within the set of integers from 1 to 27. A perfect cube is an integer that can be expressed as the cube of another integer. Let's list the perfect cubes in this range:
- 1 = 1³
- 8 = 2³
- 27 = 3³
Thus, the cube roots of these numbers (∛1, ∛8, ∛27) will be rational numbers. There are 3 favorable outcomes in our set.
Determining the Total Number of Outcomes
The total number of outcomes is simply the number of elements in the set, which is 27. Each element represents a possible outcome when selecting a number from the set.
Calculating the Probability
The probability of an event is calculated as the ratio of favorable outcomes to the total number of outcomes. In this case:
Probability (selecting a rational number) = (Number of favorable outcomes) / (Total number of outcomes)
Probability = 3 / 27
Simplifying the fraction, we get:
Probability = 1 / 9
Conclusion for Probability Calculation
Therefore, the probability of selecting a rational number from the set {∛1, ∛2, ∛3, ..., ∛27} is 1/9. This result underscores the importance of understanding the properties of numbers and how they relate to probability. This problem illustrates how mathematical concepts from different areas can be combined to solve a practical question. By carefully identifying the favorable outcomes and the total number of outcomes, we successfully calculated the desired probability. This exercise reinforces the idea that probability is a powerful tool for quantifying uncertainty and making predictions based on available information. Continuing to practice such problems will enhance your ability to analyze and solve complex scenarios involving probabilities and numerical sets.
Summary and Final Thoughts
In this article, we tackled two distinct mathematical challenges. First, we skillfully determined that sin²(100°) + cos²(80°) = 1 by applying key trigonometric identities. This process highlighted the beauty and utility of trigonometric manipulations. Then, we transitioned to a problem of probability, successfully calculating the likelihood of selecting a rational number from a set of cube roots. This exercise emphasized the crucial role of understanding number properties and applying probability principles.
These problems serve as excellent examples of how mathematical knowledge can be applied to solve diverse questions. The trigonometric problem demonstrated the power of identities in simplifying expressions, while the probability problem showcased the importance of identifying favorable outcomes and understanding set theory. Both problems underscore the value of a systematic and analytical approach to problem-solving. Mastering these fundamental concepts and techniques is essential for anyone pursuing further studies in mathematics or related fields. As you continue your mathematical journey, remember that practice and persistence are key to unlocking deeper understanding and proficiency. Embrace the challenges, explore different approaches, and celebrate the elegance and power of mathematics.
By consistently engaging with mathematical problems and concepts, you will not only strengthen your problem-solving skills but also develop a greater appreciation for the logical and interconnected nature of the mathematical world. Each problem you solve, each concept you master, will contribute to your overall mathematical fluency and confidence. So, keep exploring, keep learning, and continue to unravel the fascinating intricacies of mathematics.