Trigonometric Ratios And Reference Angles Exercise 20.3 Solutions
In this article, we will delve into solving trigonometric problems, specifically focusing on finding the values of trigonometric ratios for standard angles and determining reference angles in right-angled triangles. Trigonometry is a crucial branch of mathematics that deals with the relationships between the sides and angles of triangles. Understanding trigonometric ratios such as sine, cosine, tangent, cotangent, cosecant, and secant for specific angles, as well as the concept of reference angles, is fundamental for various applications in fields like physics, engineering, and navigation. We will tackle practical problems that involve calculating these trigonometric values and reference angles, thereby enhancing our grasp of these core concepts.
Finding the Values of Trigonometric Ratios
The first part of this exercise involves determining the values of trigonometric ratios for specific angles. These standard angles, such as 0°, 30°, 45°, 60°, and 90°, are frequently encountered in trigonometric problems. Knowing the exact values of these ratios for these angles is crucial for simplifying calculations and solving more complex problems. Each trigonometric ratio represents a specific relationship between the sides of a right-angled triangle concerning a particular angle. For instance, the sine (sin) of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine (cos) is the ratio of the adjacent side to the hypotenuse. Tangent (tan) is the ratio of the opposite side to the adjacent side. The reciprocals of these ratios – cosecant (csc), secant (sec), and cotangent (cot) – are also essential. By understanding and memorizing these fundamental trigonometric values, we can efficiently solve a wide range of trigonometric problems. This section will systematically address each given trigonometric ratio and provide a step-by-step explanation of how to determine its value.
(a) sin 90°
To find the value of sin 90°, we need to recall the definition of sine in the context of a unit circle. In a unit circle, the sine of an angle is represented by the y-coordinate of the point where the terminal side of the angle intersects the circle. For a 90° angle, this point is located at (0, 1). The y-coordinate at this point is 1. Therefore, sin 90° = 1. This is a fundamental value in trigonometry and is often used as a reference point for other trigonometric calculations. The sine function reaches its maximum value of 1 at 90°, indicating that the opposite side of the angle is equal in length to the hypotenuse in a right-angled triangle scenario. Understanding this concept is crucial for applications in physics, such as analyzing simple harmonic motion or wave behavior, where the sine function describes oscillations and periodic movements. In practical terms, knowing that sin 90° = 1 allows engineers and scientists to accurately model and predict the behavior of systems that involve angular displacement and sinusoidal patterns. It serves as a cornerstone in the broader field of trigonometry and its real-world applications. Moreover, visualizing the unit circle and understanding the geometric interpretation of sine helps in comprehending the periodic nature of trigonometric functions and their behavior across different quadrants.
(b) cos 60°
The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle, or the x-coordinate of the point on the unit circle corresponding to the angle. For cos 60°, we can consider a 30-60-90 triangle, which is a special right triangle with known side ratios. In such a triangle, the side lengths are in the ratio 1 : √3 : 2, where the shortest side (opposite the 30° angle) is 1, the longer leg (opposite the 60° angle) is √3, and the hypotenuse is 2. For a 60° angle, the adjacent side is 1, and the hypotenuse is 2. Therefore, cos 60° = adjacent / hypotenuse = 1 / 2. This value is a common one in trigonometric problems and is essential for calculations involving angles in geometry and physics. The cosine function is particularly relevant in analyzing the horizontal component of vectors or forces. For example, in mechanics, if a force is applied at a 60° angle, the horizontal component of that force is proportional to cos 60°. Understanding that cos 60° = 1/2 simplifies these types of calculations and provides a clear understanding of how forces and vectors interact. Additionally, the cosine function is used extensively in signal processing and electrical engineering to analyze alternating current (AC) circuits and wave behavior.
(c) tan 30°
The tangent of an angle is the ratio of the opposite side to the adjacent side in a right-angled triangle. To find tan 30°, we again refer to the 30-60-90 triangle. As mentioned earlier, the side lengths are in the ratio 1 : √3 : 2. For a 30° angle, the opposite side is 1, and the adjacent side is √3. Therefore, tan 30° = opposite / adjacent = 1 / √3. It is common practice to rationalize the denominator, which means multiplying both the numerator and the denominator by √3. This gives us tan 30° = (1 * √3) / (√3 * √3) = √3 / 3. This value is frequently used in problems involving slopes and angles of inclination. The tangent function is crucial in trigonometry for determining the steepness of a line or the angle at which a line rises or falls. In practical applications, the tangent function is essential in fields such as surveying and navigation, where angles and distances are used to determine positions and directions. For instance, in civil engineering, the tangent function helps calculate the slope of roads and ramps, ensuring they meet safety and functionality standards. Moreover, understanding tan 30° and its value is vital for solving problems related to vector components and force analysis in physics and engineering.
(d) cot 45°
Cotangent is the reciprocal of the tangent function. Therefore, cot θ = 1 / tan θ. To find cot 45°, we first need to determine tan 45°. For a 45-45-90 triangle, which is an isosceles right triangle, the two legs are of equal length. If we consider the two legs to be of length 1, then the hypotenuse is √2 (by the Pythagorean theorem). Thus, tan 45° = opposite / adjacent = 1 / 1 = 1. Since cotangent is the reciprocal of tangent, cot 45° = 1 / tan 45° = 1 / 1 = 1. This value is straightforward and frequently used in various trigonometric problems. The cotangent function is particularly useful in situations where the reciprocal of the tangent is more convenient to work with, such as in certain types of trigonometric identities and equations. In practical contexts, cot 45° appears in applications involving geometry and physics, where angles of 45 degrees are prevalent. For example, in projectile motion, understanding trigonometric functions at 45-degree angles helps in analyzing the trajectory and range of a projectile. Moreover, in structural engineering, the cotangent function can be used to calculate forces and stresses in structures with symmetrical designs.
(e) csc 45°
Cosecant is the reciprocal of the sine function, meaning csc θ = 1 / sin θ. To find csc 45°, we first need to determine sin 45°. As we saw in the context of cot 45°, in a 45-45-90 triangle, the legs are of equal length, and the hypotenuse is √2 times the length of a leg. If we consider the legs to be of length 1, the hypotenuse is √2. Thus, sin 45° = opposite / hypotenuse = 1 / √2. To rationalize the denominator, we multiply both the numerator and the denominator by √2, which gives us sin 45° = (1 * √2) / (√2 * √2) = √2 / 2. Now, since csc 45° = 1 / sin 45°, we have csc 45° = 1 / (√2 / 2) = 2 / √2. Again, we rationalize the denominator by multiplying both the numerator and denominator by √2, resulting in csc 45° = (2 * √2) / (√2 * √2) = 2√2 / 2 = √2. This value is another essential trigonometric result, especially useful in problems where sine reciprocals simplify the calculations. Cosecant is particularly useful in fields such as navigation and surveying, where knowing the hypotenuse relative to the opposite side is crucial. In electrical engineering, the cosecant function is used in analyzing waveforms and oscillations, particularly in situations where the reciprocal of sine simplifies the analysis of signals and their properties.
Finding the Value of Reference Angles
The second part of this exercise focuses on finding the reference angles for right-angled triangles. A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It is always an angle between 0° and 90°. Reference angles are essential because they allow us to determine the trigonometric values of any angle by relating them to angles in the first quadrant, where all trigonometric functions are positive. The process of finding reference angles involves understanding how the sides of a right-angled triangle relate to the given angles and using trigonometric ratios to determine the angle itself. Given the lengths of the sides, we can use the inverse trigonometric functions (arcsin, arccos, arctan) to find the angles. This section will systematically address the problems of finding reference angles given the side lengths of right-angled triangles, providing a clear and step-by-step approach to solving each problem.
(a) p = 4 cm and b = 4√3 cm
Here, we are given the lengths of two sides of a right-angled triangle: the perpendicular (p) is 4 cm, and the base (b) is 4√3 cm. To find the reference angle, we can use the tangent function, which relates the opposite side (perpendicular) to the adjacent side (base). The tangent of the angle θ is given by tan θ = p / b = 4 / (4√3) = 1 / √3. To find the angle θ, we take the inverse tangent (arctan) of 1 / √3. The arctan(1 / √3) is 30°. Therefore, the reference angle is 30°. This result aligns with our earlier discussion about the 30-60-90 triangle, where we established that tan 30° = 1 / √3. Understanding how to use inverse trigonometric functions is crucial for solving a wide range of problems in geometry, physics, and engineering. In fields such as surveying and navigation, determining angles from side lengths is a fundamental task. For example, if a surveyor knows the vertical and horizontal distances, they can use the arctangent function to find the angle of elevation or depression. Additionally, in mechanics, the arctangent function is used to find the angle of a force vector or the angle of a ramp or incline. The ability to calculate reference angles quickly and accurately is a valuable skill for anyone working with trigonometric concepts.
(b) p = ...
Unfortunately, the details for problem (b) are incomplete in the provided text. To accurately find the reference angle for part (b), we need the complete information, specifically the lengths of the sides of the right-angled triangle. Once we have the values for either two sides (e.g., perpendicular and base, perpendicular and hypotenuse, or base and hypotenuse) or one side and an angle, we can use trigonometric ratios and inverse trigonometric functions to determine the reference angle. If we have the lengths of the perpendicular and base, we can use the arctangent function, as demonstrated in part (a). If we have the perpendicular and hypotenuse, we can use the arcsine function (sin⁻¹(p/h)). If we have the base and hypotenuse, we can use the arccosine function (cos⁻¹(b/h)). Once the necessary information is provided, we can proceed with a similar step-by-step approach to calculate the reference angle, ensuring the solution is accurate and clearly explained. Providing the complete problem statement is crucial for delivering a comprehensive and helpful solution. Incomplete information makes it impossible to offer a specific answer, but the general method for solving such problems has been thoroughly outlined above.
In summary, we have successfully calculated the trigonometric ratios for various standard angles and determined the reference angle for a given right-angled triangle. Understanding trigonometric ratios and reference angles is fundamental for solving problems in mathematics, physics, engineering, and other scientific disciplines. The ability to find these values quickly and accurately is a valuable skill for anyone working with trigonometric concepts. By mastering these basic principles, one can tackle more complex problems involving trigonometric functions and their applications in real-world scenarios. The trigonometric ratios, such as sine, cosine, tangent, and their reciprocals, are essential tools for understanding the relationships between the sides and angles of triangles. The use of reference angles simplifies the process of finding trigonometric values for any angle by relating them to angles in the first quadrant. This article has provided a comprehensive guide to calculating these values, along with practical examples and explanations. With continued practice, these concepts can be mastered, paving the way for more advanced topics in trigonometry and its applications.
