Presentation on theme: "Angles and Degree Measure"— Presentation transcript:
1Angles and Degree Measure Chapter 13 Sec 2Angles and Degree Measure
2Standard PositionAn angle in standard position has its vertex at the origin and initial side on the positive x–axis.terminal sideinitial side
3Positively Counterclockwise Angles that have a counterclockwise rotation have a positive measure.90º130180º0º270º
4Clockwise means negative Angles that have a clockwise rotation have a negative measure.– 270º– 180º0º– 130– 90º
5Unit Circle Now let’s look at angle measures 30, 150, 210, and 330. They all form a 30° angle with the x-axis, so they should all have the same sine, cosine, and tangent values…only the signs will change!The angle to the nearest x-axis is called the reference angle.All angles with the same reference angle will have the same trig values except for sign changes.150°30°180°30º30º(1, 0)(–1, 0)30º30º210°330°
6Unit Circle A unit circle is a circle with radius 1. If we have an angle between 0o and 90o in standard position. Let P(x, y) be the point of intersection. If a perpendicular segment is drawn we create a right triangle, where y is opposite θ and x is adjacent to θ.Right triangles can be formed for angles greater than 90o, simply use the reference angle.
7Radian…stillA point P(x, y) is on the unit circle if and only if its distance from the origin is 1.The radian measure of an angle is the length of the corresponding arc on the unit circle.SinceP(x, y)sα
9Example 1 a. Change 115o to radian measure in terms of π.. b. Change radian to degree measure.
1030° and 45° RadiansYou will need to know these conversions.
11Coterminal AnglesCoterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations.Since one rotation equals 360, the measures of coterminal angles differ by multiples of 360.60300=300 – 360 =420– 60
12Example 2 Find one positive and one negative coterminal angle. a. 45o 45o + 360o = 405o and 45o – 360 o = –315ob. 225o225o + 360o = 585o and 225o – 360 o = –135o
14Radius other than 1.Suppose we have a hypotenuse with a length other than 1. For our example we’ll use r as the length.In standard position r extends from the Origin to point P(x, y).
15Quadrantal AngleIf a terminal side of an angle coincides with one of the axes, the angle is called a quadrantal angle. See below for examples:A full rotation around the circle is 360o. Measures more than 360o represent multiple rotations.
16Reference AnglesTo find the values of trig functions of angles greater than 90, you will need to know how to find the measures of the reference angle.If θ in nonquadrantal, its reference angle is formed by the terminal side of the given angle and the x-axis.
17Example 1 Find the reference angle for each angle. a. 312o Since 312o is between 270o and 360o the terminal side is in fourth quad. Therefore, 360o – 312o = 48o.b. –195othe coterminal angle is 360o – 195o = 165o this put us in the second quadrant so… 180o – 165o = 15o
18Determining Sign Students Sine values are positive All 90°(0, 1)StudentsSine values are positive(csc, too)AllAll values are positive180°0°/360°(–1, 0)(1, 0)TakeTangent values are positive(cot, too)CalculusCosine values are positive(sec, too)270°(0, –1)
19Example 2Find the values of the six trigonometric functions for angle θ in standard position if a point with coordinates (–15, 20) lies on the terminal side.
20Example 3Find the values of the six trigonometric functions Suppose θ is an angle in standard position whose terminal side lies in the Quadrant III. If find the remaining five trigonometric functions of θ.