1.1 Angles and Their Measure

GoalsIn this lesson, we will:

• Describe angles.

• Convert between radian and degree measure.

• Use radians to find values involving circles.

• Convert to degree, minute and second notation using technology.

A. AnglesParts of an angle:

• Vertex: The point where the two sides of the angle meet.

• Initial Side: The starting position of the angle, typically aligned with the positive x-axis.

• Terminal Side: The position of the angle after rotation.

Standard PositionMust have two things:

• The initial side on the positive x-axis.

• The angle is measured in a counter-clockwise direction from the initial side.The sign denotes the angle's rotation direction: Positive for counter-clockwise, negative for clockwise.

B. Radians vs. DegreesThe measure of an angle is determined by the amount of rotation from the initial side to the terminal side.Types of measures:

• Radians: A radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.

Formula: θ = s/r

• Degrees: A degree is equivalent to 1/360 of a rotation of a complete revolution about the vertex.

C. Conversion Between Them

• Degrees to RadiansExample: Convert 30° to radians.Solution:To convert degrees to radians, use the formula: [ ext{Radians} = ext{Degrees} \times \frac{\pi}{180} ][ 30 \times \frac{\pi}{180} = \frac{\pi}{6} ]

Example: Convert 300° to radians.Solution:[ 300 \times \frac{\pi}{180} = \frac{5\pi}{3} ]

Example: Convert −125° to radians.Solution:[ -125 \times \frac{\pi}{180} = -\frac{25\pi}{36} ]

• Radians to DegreesExample: Convert ( \frac{\pi}{4} ) to degrees.Solution:To convert radians to degrees, use the formula:[ ext{Degrees} = ext{Radians} \times \frac{180}{\pi} ][ \frac{\pi}{4} \times \frac{180}{\pi} = 45° ]

Example: Convert ( \frac{5\pi}{6} ) to degrees.Solution:[ \frac{5\pi}{6} \times \frac{180}{\pi} = 150° ]

Example: Convert −( \frac{\pi}{3} ) to degrees.Solution:[ -\frac{\pi}{3} \times \frac{180}{\pi} = -60° ]

Example: Convert 52 radians to degrees.Solution:[ 52 \times \frac{180}{\pi} \approx 2987.7° ]

D. Arc Length and Area of SectorThe arc length is the distance from a point on the circle to another point along the circle.Formula for arc length:[ L = r\theta \quad (\text{in radians}) ]Example: Find arc length for r = 5 and θ = 15°. Convert 15° to radians first.Conversion:[ 15 \times \frac{\pi}{180} = \frac{\pi}{12} ]Calculate arc length:[ L = 5 \times \frac{\pi}{12} = \frac{5\pi}{12} \approx 1.31 ]

Area of Sector Formula:[ A = \frac{1}{2}r²\theta \quad (\text{in radians}) ]Example: Find area for r = 5 and θ = 15° (using radians from previous calculation).Calculate area:[ A = \frac{1}{2} \times 5^2 \times \frac{\pi}{12} = \frac{25\pi}{24} \approx 3.27 ]

E. Common AnglesLet θ be an angle.(angle types)

• Acute

• Right

• Obtuse

• Straight

Degrees, Minutes, and Seconds (DMS)Degrees can be subdivided for precision.Notation:

• One minute, denoted by (′), is 1/60 of a degree.

• One second, denoted by (″), is 1/60 of a minute or 1/3600 of a degree.Examples:50 degrees, 21 minutes, and 45 seconds is written as 50°21′45″.Convert the following into decimals:Example: 50°21′45″Solution:[ 50 + \frac{21}{60} + \frac{45}{3600} \approx 50.3625° ]

Example: 115°5′20″Solution:[ 115 + \frac{5}{60} + \frac{20}{3600} \approx 115.0889° ]

Example: 32°15′55″Solution:[ 32 + \frac{15}{60} + \frac{55}{3600} \approx 32.2653° ]

Convert decimals into degrees, minutes, and seconds:Example: 40.32°Solution:

• Degrees = 40

• Minutes = [ 0.32 \times 60 \approx 19.2 ]

• Seconds = [ 0.2 \times 60 = 12 ]Result: 40°19′12″

Example: 19.99°Solution:

• Degrees = 19

• Minutes = [ 0.99 \times 60 \approx 59.4 ]

• Seconds = [ 0.4 \times 60 = 24 ]Result: 19°59′24″

Example: 61.24°Solution:

• Degrees = 61

• Minutes = [ 0.24 \times 60 \approx 14.4 ]

• Seconds = [ 0.4 \times 60 = 24 ]Result: 61°14′24″