Evaluate Trigonometric Expression Tan(cos^-1(4/5) + Sin^-1(1))
In the realm of mathematics, trigonometric functions and their inverses play a pivotal role. This article delves into the intricate details of evaluating the expression tan(cos^-1(4/5) + sin^-1(1)). We will embark on a step-by-step journey, unraveling the underlying concepts and applying relevant trigonometric identities to arrive at a solution. Mastering these kinds of problems not only enhances your understanding of trigonometry but also sharpens your problem-solving skills, crucial for various mathematical applications and competitive examinations. The fusion of inverse trigonometric functions and the tangent function provides a rich landscape for exploration, offering insights into the fundamental relationships between angles and ratios.
Understanding Inverse Trigonometric Functions
To begin, let's demystify the concept of inverse trigonometric functions. Inverse trigonometric functions, also known as arc functions, are the inverses of the basic trigonometric functions—sine, cosine, and tangent. They essentially 'undo' what the trigonometric functions do. For instance, sin^-1(x), also written as arcsin(x), gives the angle whose sine is x. Similarly, cos^-1(x), or arccos(x), yields the angle whose cosine is x, and tan^-1(x), or arctan(x), provides the angle whose tangent is x. These functions are invaluable tools in solving trigonometric equations and problems involving angles and side lengths of triangles.
Cos^-1(x): The Inverse Cosine Function
The inverse cosine function, denoted as cos^-1(x) or arccos(x), takes a value between -1 and 1 as input and returns an angle whose cosine is that value. The range of cos^-1(x) is [0, π] radians, or [0°, 180°]. This means that the output angle will always lie within this range. To visualize this, consider the unit circle. The cosine of an angle corresponds to the x-coordinate of the point on the unit circle. Therefore, cos^-1(x) seeks the angle within the upper half of the unit circle (where y-coordinates are non-negative) whose x-coordinate is x. Understanding this range is crucial for correctly interpreting the results of inverse cosine calculations and avoiding ambiguity in solutions.
Sin^-1(x): The Inverse Sine Function
Conversely, the inverse sine function, denoted as sin^-1(x) or arcsin(x), also takes a value between -1 and 1 as input but returns an angle whose sine is that value. The range of sin^-1(x) is [-π/2, π/2] radians, or [-90°, 90°]. In the context of the unit circle, the sine of an angle corresponds to the y-coordinate of the point. Consequently, sin^-1(x) seeks the angle within the right half of the unit circle (where x-coordinates are non-negative) whose y-coordinate is x. The restricted range ensures that the inverse sine function is well-defined, meaning it produces a unique output for each input within its domain. This is essential for the function to truly be an inverse of the sine function.
Evaluating cos^-1(4/5)
Now, let's apply this understanding to our specific problem. We begin by evaluating cos^-1(4/5). This means we are looking for an angle, let's call it α, such that cos(α) = 4/5. To visualize this, we can consider a right-angled triangle where the adjacent side is 4 units and the hypotenuse is 5 units. Using the Pythagorean theorem, we can find the length of the opposite side:
Opposite side = √(Hypotenuse^2 - Adjacent^2) = √(5^2 - 4^2) = √(25 - 16) = √9 = 3
Therefore, we have a right-angled triangle with sides 3, 4, and 5. Now, we can determine sin(α):
sin(α) = Opposite side / Hypotenuse = 3/5
So, if cos(α) = 4/5, then sin(α) = 3/5. This relationship is vital for later calculations when we need to apply trigonometric identities. The angle α is the angle whose cosine is 4/5, and we have also found its sine value. This approach of using right-angled triangles provides a visual and intuitive way to understand inverse trigonometric functions and their relationships.
Evaluating sin^-1(1)
Next, let's evaluate sin^-1(1). This means we are looking for an angle, let's call it β, such that sin(β) = 1. Recalling the unit circle, we know that the sine function corresponds to the y-coordinate. The y-coordinate is 1 at the angle π/2 radians (or 90°). Therefore, sin^-1(1) = π/2. This is a fundamental value that is crucial to remember. It often appears in various trigonometric problems and simplifications. Understanding the unit circle and the key angles where trigonometric functions attain specific values is essential for efficient problem-solving.
Applying the Tangent Addition Formula
With cos^-1(4/5) = α and sin^-1(1) = π/2, our original expression becomes tan(α + π/2). To evaluate this, we will use the tangent addition formula:
tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
In our case, A = α and B = π/2. So, we have:
tan(α + π/2) = (tan(α) + tan(π/2)) / (1 - tan(α)tan(π/2))
We already know sin(α) = 3/5 and cos(α) = 4/5. Therefore, we can find tan(α):
tan(α) = sin(α) / cos(α) = (3/5) / (4/5) = 3/4
Now, we also know that tan(π/2) is undefined, as the tangent function approaches infinity at π/2. This means that the direct application of the tangent addition formula will lead to an indeterminate form. However, we can use an alternative approach to handle this situation. Instead of directly substituting tan(π/2), we can use the cofunction identity to simplify the expression further.
Utilizing Cofunction Identities
To circumvent the issue of tan(π/2) being undefined, we employ cofunction identities. These identities relate trigonometric functions of complementary angles (angles that add up to π/2). Specifically, we can use the identity:
tan(π/2 + α) = -cot(α)
Cotangent, denoted as cot(α), is the reciprocal of the tangent function, i.e., cot(α) = 1/tan(α). Since we already know tan(α) = 3/4, we can find cot(α):
cot(α) = 1 / tan(α) = 1 / (3/4) = 4/3
Therefore, tan(α + π/2) = -cot(α) = -4/3. This elegant solution avoids the complexities associated with dealing with the undefined value of tan(π/2). Cofunction identities are powerful tools in trigonometry, allowing us to simplify expressions and solve problems more efficiently.
Final Solution
Thus, we have successfully evaluated the expression tan(cos^-1(4/5) + sin^-1(1)). By breaking down the problem into manageable steps, understanding the properties of inverse trigonometric functions, and applying relevant trigonometric identities, we arrived at the solution.
Therefore, tan(cos^-1(4/5) + sin^-1(1)) = -4/3. This result showcases the power of trigonometric techniques in solving complex expressions and highlights the importance of mastering fundamental concepts and identities. The journey from the initial expression to the final answer involved a combination of algebraic manipulation, trigonometric function evaluations, and the strategic application of identities. This process underscores the interconnectedness of different mathematical concepts and the need for a holistic approach to problem-solving.
In summary, this exploration not only provides the solution to the specific problem but also reinforces the underlying principles of trigonometry, making it a valuable exercise for students and enthusiasts alike. The ability to tackle such problems is a testament to one's understanding of the core concepts and their application in a cohesive manner.
