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Chapter 5: Problem 2

Indicate each angle on a circle: \(30^{\circ}, 315^{\circ},-135^{\circ}, 80^{\circ}, \frac{7 \pi}{6}, \frac{3 \pi}{4}\)

Short Answer

Expert verified
Angles are: 30掳 (Q1), 80掳 (Q1), 315掳 (Q4), -135掳 (Q3), \(\frac{7\pi}{6}\): 210掳 (Q3), \(\frac{3\pi}{4}\): 135掳 (Q2).

Step by step solution

01

Identifying Positive Angles

We first focus on the angles given in degrees that are positive: - For 30掳, it is located in the first quadrant. - For 80掳, it is also located in the first quadrant. - For 315掳, it is in the fourth quadrant, which is 45掳 less than a full circle (360掳 - 315掳 = 45掳). Identify these angles on the circumference of the circle.
02

Dealing with Negative Angles

Next, we address the negative angle of -135掳. Starting from 0掳 and moving in the clockwise direction (since it's negative), -135掳 terminates in the third quadrant equivalent to 225掳 (360掳 - 135掳 = 225掳 if moving counter-clockwise). Identify this angle on the circle.
03

Converting Radian Angles to Degrees

Convert the radian measures to degree measures to locate them on the circle:- For \(\frac{7\pi}{6}\), use the formula \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\) to get 210掳.- For \(\frac{3\pi}{4}\), convert to degrees using the same formula to get 135掳.
04

Locating Radian Angles on the Circle

Now, using the degree conversions, locate these angles on the circle:- 210掳 (from \(\frac{7\pi}{6}\)) falls in the third quadrant.- 135掳 (from \(\frac{3\pi}{4}\)) falls in the second quadrant.
05

Verifying All Angles

Verify that each calculated position for the angles is accurate by cross-referencing with known quadrant locations:- 30掳 and 80掳 are in the first quadrant.- 135掳 (both from the degree measure and \(\frac{3\pi}{4}\)) is in the second quadrant.- 210掳 (from \(\frac{7\pi}{6}\)) and -135掳 each align in the third quadrant equivalently.- 315掳 is in the fourth quadrant. Confirm each position accurately aligns with these descriptions.

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Key Concepts

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

Quadrants
When measuring angles on a circle, the circle is divided into four sections known as quadrants. These quadrants help identify the position of angles:
  • The first quadrant encompasses angles between 0掳 and 90掳, where both sine and cosine values are positive.
  • The second quadrant holds angles from 90掳 to 180掳, where sine is positive, but cosine is negative.
  • In the third quadrant, angles range from 180掳 to 270掳, with both sine and cosine values being negative.
  • The fourth quadrant contains angles from 270掳 to 360掳, where sine is negative and cosine is positive.
For instance, when identifying angles given in the exercise:- 30掳 and 80掳 are in the first quadrant.- 135掳 (converted from radians \( \frac{3\pi}{4} \)) lies in the second quadrant.- Both -135掳 and 210掳 (converted from radians \( \frac{7\pi}{6} \)) are found in the third quadrant.- Finally, 315掳 is located in the fourth quadrant.
Degree Conversion
Degrees are a common unit of angle measurement. Converting radian angles to degrees is crucial when interpreting standard angle measures. The conversion formula used is:\[ \text{degrees} = \text{radians} \times \frac{180}{\pi} \]This formula helps us translate radians into a more familiar unit:
  • For \( \frac{7\pi}{6} \), the angle converts to 210掳.
  • For \( \frac{3\pi}{4} \), the angle becomes 135掳.
These calculations allow us to locate these angles more easily on the circle using the degree system, confirming their quadrant positions.
Radian Conversion
Radian conversion is an essential skill in trigonometry and involves changing degrees into radians to handle calculations typical in advanced mathematics. To convert degrees to radians, use:\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]Radian measures are often used in calculus due to their natural simplification of mathematical analysis. When studying angles:
  • Radian values like \( \frac{7\pi}{6} \) translate to circle segments in terms of \(210^{\circ}\).
  • Likewise, \( \frac{3\pi}{4} \) corresponds to \(135^{\circ}\).
Understanding how to switch between these units facilitates working with different mathematical tools and concepts.
Trigonometry
Trigonometry is the branch of mathematics that explores the relationships and functions between angles and sides of triangles. Understanding quadrants and conversions form a base in trigonometry:
  • Using sine, cosine, and tangent, trigonometry extends beyond mere planar angles, moving into periodic functions describing waves and oscillatory motions.
  • Angular measurements, whether in degree or radian form, help define these trigonometric functions within each quadrant's unique properties.
In simpler cases like understanding which quadrant a specific angle resides in, this knowledge assists in determining the possible values for sine, cosine, and tangent, tools that are crucial for solving many real-life problems involving rotation and periodicity.

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Most popular questions from this chapter

Three airplanes depart SeaTac Airport. A NorthWest flight is heading in a direction \(50^{\circ}\) counterclockwise from East, an Alaska flight is heading \(115^{\circ}\) counterclockwise from East and a Delta flight is heading \(20^{\circ}\) clockwise from East. Find the location of the Northwest flight when it is 20 miles North of SeaTac. Find the location of the Alaska flight when it is 50 miles West of SeaTac. Find the location of the Delta flight when it is 30 miles East of SeaTac. [UW]

Convert the angle \(\frac{5 \pi}{6}\) from radians to degrees.

Erik's disabled sailboat is floating stationary 3 miles East and 2 miles North of Kingston. A ferry leaves Kingston heading toward Edmonds at \(12 \mathrm{mph} .\) Edmonds is 6 miles due east of Kingston. After 20 minutes the ferry turns heading due South. Ballard is 8 miles South and 1 mile West of Edmonds. Impose coordinates with Ballard as the origin. [UW] a) Find the equations for the lines along which the ferry is moving and draw in th lines. b) The sailboat has a radar scope that will detect any object within 3 miles of the sailboat. Looking down from above, as in the picture, the radar region looks lik circular disk. The boundary is the "edge" pr circle around this disc, the interio the inside of the disk, and the exterior is everything outside of the disk (i.e. outside of the circle). Give the mathematical (equation) description of the boundary, interior and exterior of the radar zone. Sketch an accurate picture of radar zone. Sketch an accurate picture of the radar zone by determining where line connecting Kingston and Edmonds would cross the radar zone. c) When does the ferry exit the radar zone? d) Where and when does the ferry exit the radar zone? e) How long does the ferry spend inside the radar zone?

Convert the angle \(\frac{11 \pi}{6}\) from radians to degrees.

Simplify each of the following to an expression involving a single trig function with no fractions. $$ \cos (t) \csc (t) $$
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