Solving Trigonometric Problems Finding Cos2θ And Evaluating Expressions
In this section, we will delve into the fascinating world of trigonometric identities and explore how they can be used to solve problems involving trigonometric ratios. Specifically, we'll focus on the double-angle formula for cosine, which will enable us to determine the value of cos2θ given the value of sinθ. Trigonometry, at its heart, is the study of the relationships between the sides and angles of triangles. The fundamental trigonometric ratios – sine, cosine, and tangent – form the cornerstone of this field, allowing us to connect angles to distances and vice versa. These ratios are not just abstract mathematical concepts; they have profound applications in various fields, including physics, engineering, navigation, and even computer graphics. Understanding these ratios and their interrelationships is crucial for tackling a wide range of problems in both theoretical and practical contexts.
The double-angle formulas are a powerful set of identities that allow us to express trigonometric functions of twice an angle in terms of trigonometric functions of the angle itself. The double-angle formula for cosine, which is the key to solving this problem, comes in three equivalent forms:
- cos2θ = cos²θ - sin²θ
- cos2θ = 2cos²θ - 1
- cos2θ = 1 - 2sin²θ
The choice of which form to use often depends on the information given in the problem. In this case, since we are given the value of sinθ, the third form (cos2θ = 1 - 2sin²θ) appears to be the most convenient. This form directly relates cos2θ to sinθ, allowing us to bypass the need to calculate cosθ separately. However, it's worth noting that we could also use the first form (cos2θ = cos²θ - sin²θ) if we first determined cosθ using the Pythagorean identity (sin²θ + cos²θ = 1). This would involve an extra step but would ultimately lead to the same result, providing a good illustration of the interconnectedness of trigonometric identities. The double-angle formulas are not just mathematical curiosities; they are essential tools in simplifying trigonometric expressions, solving equations, and evaluating integrals. They also play a critical role in more advanced topics such as Fourier analysis and signal processing.
Now, let's apply our knowledge to the problem at hand. We are given that sinθ = 12/15. Our goal is to find the value of cos2θ. As we discussed earlier, the double-angle formula cos2θ = 1 - 2sin²θ seems ideally suited for this task. We simply need to substitute the given value of sinθ into this formula and perform the necessary calculations. This direct approach highlights the elegance and efficiency of using the appropriate trigonometric identity to solve a problem. By recognizing the relationship between cos2θ and sinθ expressed in the formula, we can avoid unnecessary steps and arrive at the solution quickly. This skill of identifying and applying the correct identity is a hallmark of proficiency in trigonometry.
Substituting sinθ = 12/15 into the formula, we get:
cos2θ = 1 - 2(12/15)²
This equation now involves only arithmetic operations. We need to square the fraction 12/15, multiply the result by 2, and then subtract that value from 1. Order of operations is crucial here to ensure we arrive at the correct answer. A common mistake is to perform the subtraction before the multiplication, which would lead to an incorrect result. Careful attention to detail and a methodical approach are essential for accurate calculations in trigonometry, as in any mathematical discipline. The next step involves simplifying the expression and arriving at the final numerical value for cos2θ. This will demonstrate the practical application of the double-angle formula and solidify our understanding of its utility.
Simplifying the expression further:
cos2θ = 1 - 2(144/225) cos2θ = 1 - 288/225
To combine the terms, we need to find a common denominator, which in this case is 225. We can rewrite 1 as 225/225:
cos2θ = 225/225 - 288/225 cos2θ = -63/225
The fraction -63/225 can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 9:
cos2θ = (-63 ÷ 9) / (225 ÷ 9) cos2θ = -7/25
Therefore, the value of cos2θ is -7/25. This result demonstrates how the double-angle formula can be effectively used to determine trigonometric function values for multiple angles when the values for single angles are known. This skill is fundamental in various applications, including solving trigonometric equations and analyzing periodic phenomena.
In this section, we will tackle a problem that involves evaluating a trigonometric expression given the value of the tangent function. This type of problem often requires us to manipulate trigonometric ratios and utilize their interrelationships to arrive at the solution. The key to success here lies in recognizing that all trigonometric ratios can be expressed in terms of each other, and that the tangent function, in particular, can serve as a bridge between sine and cosine. Trigonometric expressions, like the one we are dealing with, are ubiquitous in various branches of mathematics and physics. They appear in the study of waves, oscillations, and periodic motion, as well as in geometric problems involving triangles and circles. Being able to simplify and evaluate these expressions is a crucial skill for anyone working in these areas.
We are given that tanθ = 1/2. Recall that the tangent function is defined as the ratio of sine to cosine: tanθ = sinθ/cosθ. This fundamental relationship is the cornerstone of our approach to this problem. It allows us to connect the given information about tanθ to the sine and cosine terms that appear in the expression we need to evaluate. The definition of the tangent function is not just a dry mathematical formula; it has a geometric interpretation as the slope of a line in the coordinate plane. This geometric perspective can provide valuable intuition and help us visualize trigonometric relationships more clearly. Moreover, the tangent function is intimately related to the concept of angles of inclination, which are crucial in fields such as surveying and navigation. Therefore, a deep understanding of the tangent function is essential for both theoretical and practical applications of trigonometry.
Our goal is to evaluate the expression (2cosθ + 5sinθ) / (cosθ + 2sinθ). This expression involves both sine and cosine terms, but it doesn't directly involve the tangent function. To make use of the given information (tanθ = 1/2), we need to find a way to introduce tanθ into the expression. A common technique for achieving this is to divide both the numerator and denominator of the expression by either cosθ or sinθ. The choice of which to divide by depends on the specific problem and the desired outcome. In this case, dividing both the numerator and denominator by cosθ seems promising because it will directly introduce the ratio sinθ/cosθ, which is equal to tanθ. This manipulation is a powerful tool in simplifying trigonometric expressions. It allows us to rewrite expressions in a more convenient form, often revealing hidden relationships and making evaluation easier. The act of dividing both the numerator and denominator by the same quantity is a fundamental algebraic technique that preserves the value of the expression while changing its appearance. This technique is widely used in various mathematical contexts, and its application in trigonometry is a testament to its versatility.
Dividing both the numerator and the denominator by cosθ, we get:
[(2cosθ + 5sinθ) / cosθ] / [(cosθ + 2sinθ) / cosθ]
Now, we distribute the division in both the numerator and the denominator:
(2cosθ/cosθ + 5sinθ/cosθ) / (cosθ/cosθ + 2sinθ/cosθ)
This step involves applying the distributive property of division over addition, a basic algebraic principle that is essential for manipulating fractions and expressions. The careful application of this property ensures that each term in the numerator and denominator is correctly divided by cosθ, setting the stage for the introduction of the tangent function. The distributive property is not just a mathematical rule; it reflects the fundamental way in which operations interact with each other. Its use in this context highlights the importance of understanding basic algebraic principles in tackling trigonometric problems.
Simplifying the fractions, we have:
(2 + 5tanθ) / (1 + 2tanθ)
We have successfully transformed the original expression into one that involves tanθ. This was the key step in solving this problem. By strategically dividing both the numerator and denominator by cosθ, we were able to leverage the given information about tanθ and rewrite the expression in a form that is readily evaluated. This technique demonstrates the power of algebraic manipulation in simplifying trigonometric expressions and revealing their underlying structure. The ability to recognize and execute such manipulations is a hallmark of mathematical proficiency. Now, we can substitute the given value of tanθ into this simplified expression and calculate the final result.
Now, substitute tanθ = 1/2:
(2 + 5(1/2)) / (1 + 2(1/2))
This is now a simple arithmetic problem. We need to perform the multiplications and additions in the correct order to arrive at the final answer. The order of operations is a fundamental principle in mathematics that dictates the sequence in which operations should be performed to ensure consistent and accurate results. In this case, we first perform the multiplications (5(1/2) and 2(1/2)) and then the additions. Careful attention to the order of operations is crucial for avoiding errors and arriving at the correct solution.
Simplifying further:
(2 + 5/2) / (1 + 1) (4/2 + 5/2) / 2 (9/2) / 2
To divide a fraction by a whole number, we can multiply the denominator of the fraction by the whole number:
9/4
Therefore, the value of the expression (2cosθ + 5sinθ) / (cosθ + 2sinθ) when tanθ = 1/2 is 9/4. This result showcases the power of trigonometric identities and algebraic manipulation in solving problems involving trigonometric ratios. By strategically applying these techniques, we were able to transform a complex expression into a simple numerical value.
