Solving For X In Cos(x) = Sin(14) Where 0 < X < 90
In the realm of trigonometry, solving equations involving trigonometric functions is a common task. Here, we delve into solving a specific trigonometric equation where we aim to find the value of x that satisfies the condition cos(x) = sin(14°), with the constraint that x lies between 0° and 90°. This problem utilizes the fundamental relationship between sine and cosine functions within a right-angled triangle and their complementary angles. To effectively solve this, we will explore the properties of trigonometric functions and apply the relevant identities to arrive at the correct solution. Understanding these concepts is crucial not only for solving this particular problem but also for tackling a wide array of trigonometric problems.
Understanding Trigonometric Relationships
Before diving into the solution, it's important to grasp the fundamental trigonometric relationships. Sine and cosine are two primary trigonometric functions that describe the relationship between angles and sides in a right-angled triangle. Specifically, in a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. A crucial relationship to remember is the complementary angle identity, which states that the sine of an angle is equal to the cosine of its complement, and vice versa. Mathematically, this is expressed as:
sin(θ) = cos(90° - θ)
cos(θ) = sin(90° - θ)
These relationships are derived from the geometry of a right-angled triangle. If one acute angle in the triangle is θ, the other acute angle is 90° - θ, and the side opposite θ is adjacent to 90° - θ, and vice versa. This understanding of complementary angles and their trigonometric functions is key to solving our equation. Moreover, within the range of 0° to 90°, sine and cosine functions exhibit specific behaviors; sine increases from 0 to 1, while cosine decreases from 1 to 0. This knowledge helps in verifying the solution and understanding the context of the problem.
Applying the Complementary Angle Identity
Our goal is to find the value of x for which cos(x) = sin(14°), where 0° < x < 90°. Recognizing the complementary angle identity, we can rewrite the sine function in terms of cosine. We know that sin(θ) = cos(90° - θ). Applying this to our equation, we can rewrite sin(14°) as:
sin(14°) = cos(90° - 14°)
sin(14°) = cos(76°)
Now, our original equation cos(x) = sin(14°) can be rewritten as:
cos(x) = cos(76°)
This transformation is crucial because it allows us to directly compare the cosine functions on both sides of the equation. By expressing both sides in terms of cosine, we can equate the angles, leading us closer to the solution. The complementary angle identity serves as a bridge, connecting sine and cosine functions and simplifying the problem. Understanding when and how to apply these identities is a fundamental skill in trigonometry and allows for efficient problem-solving.
Solving for x
Now that we have the equation cos(x) = cos(76°), we can directly compare the angles. Since the cosine function yields the same value for two angles within the range of 0° to 90° only if the angles are equal, we can deduce that:
x = 76°
This is a straightforward deduction, but it's essential to remember the context of the problem. We were given the constraint that 0° < x < 90°. Our solution, x = 76°, falls within this range, confirming that it is a valid solution. Had the solution fallen outside this range, we would need to consider other possibilities, such as using the periodicity or symmetry properties of trigonometric functions to find an equivalent angle within the desired range. In this case, however, the direct comparison provides a clear and correct solution. It is always crucial to check the solution against the given constraints to ensure its validity.
Conclusion
In conclusion, by utilizing the complementary angle identity and the fundamental properties of trigonometric functions, we have successfully found the value of x that satisfies the equation cos(x) = sin(14°), with the constraint that 0° < x < 90°. The solution is:
x = 76°
This problem underscores the importance of understanding and applying trigonometric identities. The complementary angle identity is a powerful tool that allows us to relate sine and cosine functions, simplifying complex trigonometric equations. Moreover, this exercise reinforces the significance of working within specified constraints and verifying solutions to ensure their validity. Trigonometry is a vital branch of mathematics with applications in various fields, including physics, engineering, and navigation. Mastering the fundamental concepts and techniques, such as the ones demonstrated here, is essential for further exploration of mathematical and scientific principles. This problem serves as a valuable learning experience, highlighting the interconnectedness of trigonometric concepts and the importance of a systematic approach to problem-solving. By breaking down the problem into smaller steps and applying the appropriate identities, we arrived at a clear and accurate solution.
Therefore, the correct answer is D. 76°
