Problem 52
Question
find the reference angle for each angle. $$ 553^{\circ} $$
Step-by-Step Solution
Verified Answer
The reference angle for \(553^{\circ}\) is \(13^{\circ}\).
1Step 1: Reduce the angle
Since \(360^{\circ}\) represent a complete revolution, angle greater than \(360^{\circ}\) can be reduced by subtracting \(360^{\circ}\) multiple times until the angle falls within range of \(0^{\circ}\) to \(360^{\circ}\). So, \(553^{\circ} - 360^{\circ} = 193^{\circ}\)
2Step 2: Find the reference angle
The reference angles are always positive and measured counter-clockwise. Because the result from step 1, \(193^{\circ}\), in the second quadrant, its reference angle can be found by subtracting it from \(180^{\circ}\):\(180^{\circ} - 193^{\circ} = -13^{\circ}\)The reference angle is always positive so we take the absolute value:\(|-13^{\circ}| = 13^{\circ}\)
Key Concepts
Angle ReductionTrigonometric FunctionsQuadrantal Angles
Angle Reduction
Angle reduction is a method used to simplify the computation of trigonometric functions for angles larger than one full rotation, or greater than \(360^{\text{o}}\). The process involves subtracting \(360^{\text{o}}\) from the given angle until the resulting angle is less than or equal to \(360^{\text{o}}\). This is necessary because the trigonometric functions are periodic, meaning they repeat their values in regular intervals. For our example, we start with an angle of \(553^{\text{o}}\).
By reducing this angle, we subtract \(360^{\text{o}}\) which is one full rotation: \(553^{\text{o}} - 360^{\text{o}} = 193^{\text{o}}\). Now that the angle is within the first full circle, or less than \(360^{\text{o}}\), we can find the related trigonometric function values more easily. Remember, the goal of angle reduction is not to change the essence of the angle's position or its trigonometric function values but simply to work with it within the principal range of \(0^{\text{o}}\) to \(360^{\text{o}}\).
Practicality-wise, reducing angles can help in a wide range of applications, from solving geometry problems to simplifying complex calculations in physics and engineering.
By reducing this angle, we subtract \(360^{\text{o}}\) which is one full rotation: \(553^{\text{o}} - 360^{\text{o}} = 193^{\text{o}}\). Now that the angle is within the first full circle, or less than \(360^{\text{o}}\), we can find the related trigonometric function values more easily. Remember, the goal of angle reduction is not to change the essence of the angle's position or its trigonometric function values but simply to work with it within the principal range of \(0^{\text{o}}\) to \(360^{\text{o}}\).
Practicality-wise, reducing angles can help in a wide range of applications, from solving geometry problems to simplifying complex calculations in physics and engineering.
Trigonometric Functions
Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in the study of periodic phenomena, such as sound and light waves, and are crucial in various scientific and engineering fields. The primary trigonometric functions are sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent.
Each of these functions has a specific definition and property but what they all share in common is their periodic nature. For instance, the sine and cosine functions have a period of \(360^{\text{o}}\), which means they repeat their values every \(360^{\text{o}}\) of rotation. Understanding the periodicity of these functions helps in angle reduction, since an angle can be reduced to a more manageable size while preserving the values of trigonometric functions.
Moreover, knowledge of these functions also aids in finding reference angles. For an angle in any quadrant, you can use its sine, cosine, or tangent value to easily determine the measure of its reference angle, which will always be a value between \(0^{\text{o}}\) and \(90^{\text{o}}\), and has the same sine, cosine, and tangent values as the original angle when considering sign.
Each of these functions has a specific definition and property but what they all share in common is their periodic nature. For instance, the sine and cosine functions have a period of \(360^{\text{o}}\), which means they repeat their values every \(360^{\text{o}}\) of rotation. Understanding the periodicity of these functions helps in angle reduction, since an angle can be reduced to a more manageable size while preserving the values of trigonometric functions.
Moreover, knowledge of these functions also aids in finding reference angles. For an angle in any quadrant, you can use its sine, cosine, or tangent value to easily determine the measure of its reference angle, which will always be a value between \(0^{\text{o}}\) and \(90^{\text{o}}\), and has the same sine, cosine, and tangent values as the original angle when considering sign.
Quadrantal Angles
Quadrantal angles are those angles that lie precisely on the x-axis or y-axis when plotted on the coordinate plane. They are the angles of \(0^{\text{o}}\), \(90^{\text{o}}\), \(180^{\text{o}}\), and \(270^{\text{o}}\)—each representing a right angle's multiple. In terms of trigonometric functions, quadrantal angles lead to values that are either 0, 1, or -1, which simplifies the computation significantly.
When finding reference angles, particularly, the quadrantal angles play a significant role. The reference angle is the smallest angle that the given angle makes with the x-axis, and it is always measured in the positive direction, counter-clockwise from the x-axis. For non-quadrantal angles, like our example of \(193^{\text{o}}\), which falls in the second quadrant, the reference angle can be found by determining how far the angle is from the nearest x-axis. In this case, since \(193^{\text{o}}\) is past \(180^{\text{o}}\), we subtract \(193^{\text{o}}\) from \(180^{\text{o}}\) to get \(-13^{\text{o}}\), and then take the absolute value to find the reference angle of \(13^{\text{o}}\).
Understanding quadrantal angles is critical for efficiently solving trigonometric problems, predicting the signs of trigonometric functions, and simplifying calculations as they provide a clear frame of reference in the unit circle.
When finding reference angles, particularly, the quadrantal angles play a significant role. The reference angle is the smallest angle that the given angle makes with the x-axis, and it is always measured in the positive direction, counter-clockwise from the x-axis. For non-quadrantal angles, like our example of \(193^{\text{o}}\), which falls in the second quadrant, the reference angle can be found by determining how far the angle is from the nearest x-axis. In this case, since \(193^{\text{o}}\) is past \(180^{\text{o}}\), we subtract \(193^{\text{o}}\) from \(180^{\text{o}}\) to get \(-13^{\text{o}}\), and then take the absolute value to find the reference angle of \(13^{\text{o}}\).
Understanding quadrantal angles is critical for efficiently solving trigonometric problems, predicting the signs of trigonometric functions, and simplifying calculations as they provide a clear frame of reference in the unit circle.
Other exercises in this chapter
Problem 52
Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \cos (2 \pi x+4 \pi)$$
View solution Problem 52
In Exercises 45–52, graph two periods of each function. $$ y=\left|\tan \frac{1}{2} x\right| $$
View solution Problem 53
Use a sketch to find the exact value of each expression. $$ \sin \left(\cos ^{-1} \frac{\sqrt{2}}{2}\right) $$
View solution Problem 53
A forest ranger sights a fire directly to the south. A second ranger, 7 miles east of the first ranger, also sights the fire. The bearing from the second ranger
View solution