5.3 Angles and Their Measure

Transcription

1 5.3 Angles and Their Measure 1. Angles and their measure 1.1. Angles. An angle is formed b rotating a ra about its endpoint. The starting position of the ra is called the initial side and the final position of the ra is the terminal side. The common endpoint V of the initial side and terminal side is called the vertex. A counterclockwise rotation produces a positive angle, and a clockwise rotation produces a negative angle. See Figure 1(a)-(b), where the Greek letter θ has been used to denote an angle. Two different angles ma have the same initial and terminal sides. Such angles are said to be coterminal angles (Figure 2). Terminal Side Vertex Initial Side θ Terminal side θ θ Vertex Initial Side Vertex Initial side x Terminal Side (a) (b) (c) Figure 1. (a). A Positive Angle θ, (b). A Negative angle θ, (c). Standard Position An angle in a rectangular coordinate sstem is said to be standard position if its vertex is at the origin and the initial side is along the positive x axis (Figure 1(c)). If the terminal side of an angle in standard position lies along a coordinate axis, the angle is said to be a quadrantal angle. If the terminal side does not lie along a coordinate axis, then the angle is often referred to in terms of the quadrant in which the terminal side lies. Degree and Radian Measure There are two common sstems to measure the size of an angles: degree measure and radian measure. Definition 1 (Degree Measure) An angle formed b one complete rotation is said to have a measure of 360 degrees (360 o ). An angle formed b of a complete rotation is said to have a measure of 1 degree (1o ). The smbol o denotes degrees ( see Figure 3). Certain angles have special names(figure 4): Right angle: right angle has measure 90 o. Straight angle: straight angle has measure o. Acute angle: The measure of an acute angle is greater than 0 o and less than 90 o. 1

2 2 75o 285 oo x O o 830 o o 110 o x O x Figure 2. Coterminal Angles 120 o 360 o 45o 130 o Figure 3. Degree Measure Obtuse angle: The measure of an obtuse angle is greater than 90 o but less than o. We will use the Greek letters α, β, γ and θ to denote angles. For simplicit, we sometimes refer to an angle θ having measure 45 o as a 45 o angle, or an angle of 45 o. This ma be expressed as θ = 45 o. Two positive angles are complementar angles if their sum equals 90 o and are supplementar angles is their sum is o 5. A degree can be divided further using decimal notation. Fractions of a degree ma be measured using minutes and seconds. Each degree is divided into 60 equal parts called minutes, and each minute is divided into 60 equal parts call seconds. Smbolicall, minutes are represented b and seconds b. For example, the measurement 25 o represents 25 degrees, 45 minutes, 30 second. Express in decimal degrees, this measurement is 24 o = 24 o + ( )o + ( )o = o. Example (A). Convert 21 o to decimal degrees. (B). Convert o to degree-minute-second form.

3 3 90 o o 60o 130 o Right Angle Straight Angle Acute Angle Obtuse Angle Figure 4. Tpes of Angles β = 55 o α = 35 o Complementar Angles β = 120 o α = 60 o Supplementar Angles Figure 5. Complementar angles and supplementar angles Solution. (A). 21 o = ( (B) )o = o o = 105 o + ( ) = 105 o = 105 o ( ) = 105 o = 105 o Example Find angles that are complementar and supplementar to α = 34 o

4 4 Solution. If angle β is complementar to α, then β = 90 o α. β = 90 o 34 o = 89 o o (90 o = 89 o ) = 55 o An supplementar angle to α is given b γ = o α. γ = o 34 o = 179 o o ( o = 179 o ) = 145 o Radian Measure A second unit of angle measure is radians. radian measure is a common unit of measurement in man technical, including calculus. Angle θ in Figure 6 has a measure of one radian. The vertex of θ is located at the center of the circle, and its initial and terminal sides intercept an arc whose length is equal to the radius of the circle. θ r r x Figure 6. One radian Radian measure An angle that has its vertex at the center of a circle and intercepts an arc on the circle equal in length to the radius of the circle has a measure of one radian. From geometr we know that the arc length s on a circle is proportional to the measure of the central angle θ. A central angle of 2π radians corresponds to an arc length that equals the circumference C = 2π r. Using proportions, s θ = 2 π r 2 π,

5 which simplifies to s = r θ. Hence, the radian measure of θ can also be given b And when s = r, then θ = s r θ = r r = 1. Note: s and r must be measured in the same units. Also note that θ is being used in two was: as the name of an angle and as the measure of the angle. The context determines the choice. Thus, when we write θ = s/r, we mean the radian measure of angle θ is s/r. From Degrees to Radians, and Vice Versa The circumference of a circle with radius r is C = 2π r. Therefore one rotation contains 2 π radians, and so 360 o is equivalent to 2 π radians. Radian measure can be compared to degree measure using proportions. Since o is equivalent to π radians, it follows that radian measure degree measure = Solving for radian measure results in π o. radian measure = degree measure and solving for degree measure results in π o degree measure = radian measure o π. 5 Radian-Degree Conversion Formulas θ deg = θ o rad π rad Basic proportion θ deg = o rad θ rad Radians to degrees θ rad = π rad θ o deg Degrees to radians Degrees 0 o 30 o 45 o 60 o 90 o o 360 o π π π π Radians π 2 π Table 1. Equivalents measures in degrees and radians. ExampleConvert the degree measures to radian measures, and radian measures to degree measures. (a). 75 o, (b). 5 radian, (c). 41 o 12, (d). 1 Solution. (a). θ rad = π rad θ o deg = π 75 = 5 π 12 = (b). θ deg = π rad θ rad = π 5 = 900 π = 286.5o.

6 6 (c). 41 o 12 = ( )o = 41.2 o θ rad = π rad o θ deg = π 41.2 = (d). θ deg = π rad θ rad = π 1 = 37.3o Reading: Section 5.3 Exercise 5.3: