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Chapter 4: Problem 72

Determine the angle of the smallest possible positive measure that is coterminal with each of the angles whose measure is given. Use degree or radian measures accordingly. $$-187^{\circ}$$

Short Answer

Expert verified
The smallest positive coterminal angle is \(173^{\circ}\).

Step by step solution

01

Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side. To find a positive coterminal angle, we need to add or subtract full circles (360掳) until we get a positive angle.
02

Adding 360掳 to Find a Positive Angle

Since we start with a negative angle, we add 360掳 to \[-187^{\circ}\] to find a coterminal angle:\[-187^{\circ} + 360^{\circ} = 173^{\circ}.\]
03

Verify the Positive Angle

Check if the angle \(173^{\circ}\) is positive and less than 360掳. Since it is, \(173^{\circ}\) is indeed the smallest positive coterminal angle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Angle Measurement
Angle measurement is a way to quantify the amount of rotation between two intersecting lines having a common point called the vertex. This measurement can be in degrees or radians. For the current discussion, let's focus on degree measurement as it is one of the most common methods. To visualize, think of a circular protractor divided into 360 equal parts---each part is one degree. Hence, a complete revolution in a circle is 360掳. This divides the angle measurement into:
  • Acute Angles: Less than 90掳
  • Right Angles: Exactly 90掳
  • Obtuse Angles: Between 90掳 and 180掳
  • Straight Angles: Exactly 180掳
Understanding these categories can help you easily identify coterminal angles when given a problem.
Exploring Positive Angles
Positive angles are those angles that are measured in the counterclockwise direction from the initial side. In the context of coterminal angles, when you are tasked with finding a positive angle that is coterminal with a given angle, you may need to adjust the original measurement.For instance, if you have a negative angle like \[-187^{\circ}\], this signifies a clockwise rotation. To convert this into a positive angle, you add a full circle, or 360掳. By doing so, \[-187^{\circ} + 360^{\circ} = 173^{\circ}\], getting a positive angle. This simple adjustment is crucial since coterminal angles can be both positive and negative, but the smallest positive measure is often the sought-after solution in these exercises.
Concept of Degree Measure
The degree measure is a unit of angle measurement which divides one complete rotation into 360 equal parts. This method is deeply integrated into various mathematical contexts and practical applications around the world.Degree measure plays a critical role in understanding coterminal angles. To determine if two angles, like \(360n + \theta\), where \(n\) is an integer, are coterminal, you can compare their degree measures. When working with exercises involving negative angles, converting them, as shown in previous steps, ensures one finds the smallest positive coterminal angle.By consistently applying the knowledge that 360掳 represents one full revolution, solving such problems becomes intuitive. This approach is valuable because it helps further explore the vast applications of angle relationships in geometry.

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Most popular questions from this chapter

In calculus, the value \(F(b)-F(a)\) of a function \(F(x)\) at \(x=a\) and \(x=b\) plays an important role in the calculation of definite integrals. Find the exact value of \(F(b)-F(a)\) $$F(x)=2 \tan x+\cos x, a=-\frac{\pi}{6}, b=\frac{\pi}{4}$$

Explain the mistake that is made. If the radius of a set of tires on a car is 15 inches and the tires rotate \(180^{\circ}\) per second, how fast is the car traveling (linear speed) in miles per hour? Solution: Write the formula for linear speed. $$\begin{aligned}&v=r \omega\\\&v=(15 \text { in. })\left(\frac{180^{\circ}}{\sec }\right) \end{aligned}$$ Let \(r=15\) inches and \(\omega=180^{\circ}\) per second. Simplify. \(v=2700 \frac{\mathrm{in}}{\mathrm{sec}}\) Let 1 mile \(=5280\) feet \(=63,360\) inches and 1 hour \(=3600\) seconds. $$v=\left(\frac{2700 \cdot 3600}{63,360}\right) \mathrm{mph}$$ Simplify. \(v \approx 153.4 \mathrm{mph}\) This is incorrect. The correct answer is approximately 2.7 mph. What mistake was made?

In calculus we work with real numbers; thus, the measure of an angle must be in radians. An object is rotating at \(600^{\circ}\) per second, find the central angle \(\theta,\) in radians, when \(t=3 \mathrm{sec}\).

Explain the mistake that is made. Solve the triangle \(a=6, b=2,\) and \(c=5\). Solution: Step 1: Find \(\beta\) Apply the Law of Cosines. \(b^{2}=a^{2}+c^{2}-2 a c \cos \beta\) Solve for \(\beta\) \(\beta=\cos ^{-1}\left(\frac{a^{2}+c^{2}-b^{2}}{2 a c}\right)\) Let \(a=6, b=2\) \(c=5 . \quad \beta \approx 18^{\circ}\) Step 2: Find \(\alpha\) \(\begin{array}{ll}\text { Apply the Law } & \frac{\sin \alpha}{a}=\frac{\sin \beta}{b} \\ \text { of sines. } & a\end{array}\) Solve for \(\alpha\) \(\alpha=\sin ^{-1}\left(\frac{a \sin \beta}{b}\right)\) Let \(a=6, b=2\) and \(\beta=18^{\circ}\) \(\alpha \approx 68^{\circ}\) Step 3: Find \(\gamma\) \(\alpha+\beta+\gamma=180^{\circ}\) $$ \begin{aligned} 68^{\circ}+18^{\circ}+\gamma &=180^{\circ} \\ \gamma & \approx 94^{\circ} \end{aligned} $$ \(a=6, b=2, c=5, \alpha \approx 68^{\circ}, \beta \approx 18^{\circ},\) and \(\gamma \approx 94^{\circ}\) This is incorrect. The longest side is not opposite the largest angle. What mistake was made?

Determine whether each statement is true or false. It is possible for all six trigonometric functions of the same angle to have positive values.
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