Problem 26
Question
\(25-30\) me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ 135^{\circ} $$
Step-by-Step Solution
Verified Answer
The positive coterminal angles are \(495^{\circ}\) and \(855^{\circ}\); the negative coterminal angles are \(-225^{\circ}\) and \(-585^{\circ}\).
1Step 1: Understanding Coterminal Angles
Coterminal angles are angles that share the same initial side and terminal side. To find coterminal angles, you add or subtract full rotations of the circle, which is \(360^{\circ}\).
2Step 2: Finding Positive Coterminal Angles
Start with the given angle, \(135^{\circ}\). To find two positive coterminal angles, add \(360^{\circ}\) to obtain the first: \(135^{\circ} + 360^{\circ} = 495^{\circ}\). For the second positive angle, add \(360^{\circ}\) again: \(495^{\circ} + 360^{\circ} = 855^{\circ}\).
3Step 3: Finding Negative Coterminal Angles
To find two negative coterminal angles, subtract \(360^{\circ}\) from \(135^{\circ}\). First, calculate: \(135^{\circ} - 360^{\circ} = -225^{\circ}\). For the second negative angle, subtract \(360^{\circ}\) again: \(-225^{\circ} - 360^{\circ} = -585^{\circ}\).
Key Concepts
Coterminal Angle CalculationPositive and Negative AnglesAngle Measures in Degrees
Coterminal Angle Calculation
Coterminal angles are angles that share the same terminal side in a coordinate plane. To find coterminal angles, you can add or subtract full circle rotations to the given angle. Each full rotation in degrees is exactly \(360^{\circ}\).
By using this method, you can find any number of coterminal angles. For example, to find two positive angles coterminal with \(135^{\circ}\), you add \(360^{\circ}\) twice: first, \(135^{\circ} + 360^{\circ} = 495^{\circ}\), and then \(495^{\circ} + 360^{\circ} = 855^{\circ}\).
Similarly, for negative coterminal angles, you subtract \(360^{\circ}\) repeatedly: \(135^{\circ} - 360^{\circ} = -225^{\circ}\), and then \(-225^{\circ} - 360^{\circ} = -585^{\circ}\).
This process shows how different angles can be coterminal while maintaining their orientation in relation to the positive x-axis.
By using this method, you can find any number of coterminal angles. For example, to find two positive angles coterminal with \(135^{\circ}\), you add \(360^{\circ}\) twice: first, \(135^{\circ} + 360^{\circ} = 495^{\circ}\), and then \(495^{\circ} + 360^{\circ} = 855^{\circ}\).
Similarly, for negative coterminal angles, you subtract \(360^{\circ}\) repeatedly: \(135^{\circ} - 360^{\circ} = -225^{\circ}\), and then \(-225^{\circ} - 360^{\circ} = -585^{\circ}\).
This process shows how different angles can be coterminal while maintaining their orientation in relation to the positive x-axis.
Positive and Negative Angles
Angles can be categorized as positive or negative based on the direction they are measured from the initial side. Positive angles are measured counterclockwise from the initial side, whereas negative angles are measured clockwise. This direction helps determine the angle's sign.
In the example given, the original angle \(135^{\circ}\), as well as the calculated \(495^{\circ}\) and \(855^{\circ}\), are positive angles because they are all measured counterclockwise from the positive x-axis.
On the other hand, negative coterminal angles such as \(-225^{\circ}\) and \(-585^{\circ}\) represent angles measured clockwise from the positive x-axis. Understanding these direction and measurement differences is crucial for working with trigonometric functions and their applications.
In the example given, the original angle \(135^{\circ}\), as well as the calculated \(495^{\circ}\) and \(855^{\circ}\), are positive angles because they are all measured counterclockwise from the positive x-axis.
On the other hand, negative coterminal angles such as \(-225^{\circ}\) and \(-585^{\circ}\) represent angles measured clockwise from the positive x-axis. Understanding these direction and measurement differences is crucial for working with trigonometric functions and their applications.
Angle Measures in Degrees
Angle measures can be represented in both degrees and radians, but in this context, we focus on degrees. Degrees are a more intuitive means of describing angles in terms of circle fractions, where one full circle equals \(360^{\circ}\).
It's important to recognize that angles can have equivalent measures that form multiples of \(360^{\circ}\), leading to coterminal angles without changing their geometric meaning, just the numerical representation.
When working with degree measures, it allows for straightforward calculations using simple addition and subtraction to find coterminal angles. This concept is widely used in navigation, engineering, and physics, where precise angle measurements are essential.
It's important to recognize that angles can have equivalent measures that form multiples of \(360^{\circ}\), leading to coterminal angles without changing their geometric meaning, just the numerical representation.
When working with degree measures, it allows for straightforward calculations using simple addition and subtraction to find coterminal angles. This concept is widely used in navigation, engineering, and physics, where precise angle measurements are essential.
Other exercises in this chapter
Problem 25
Evaluate the expression without using a calculator. $$\sin 30^{\circ} \cos 60^{\circ}+\sin 60^{\circ} \cos 30^{\circ}$$
View solution Problem 26
Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ b=73, \quad c=82, \quad \angle B=58^{\circ} $$
View solution Problem 26
9–32 Find the exact value of the trigonometric function. $$\tan \frac{5 \pi}{6}$$
View solution Problem 26
Evaluate the expression without using a calculator. $$\left(\sin 60^{\circ}\right)^{2}+\left(\cos 60^{\circ}\right)^{2}$$
View solution