Transcript Sec. 6.1
Sec. 6.1 Angles and Their Measures Trigonometry (Greek) – measurement of triangles • Angle – Determined by rotating a ray about its endpoint – – – – Positive angles – Rotation Counterclockwise Negative angles – Rotation clockwise Initial Side – Starting position of the ray Terminal Side – Position after the ray is rotated – Vertex – Endpoint of the ray • Standard Position – on the coordinate plane the initial side lines up on the x-axis and the vertex is at the origin • Pictures p. 454 6.1, 6.2 • Greek letters denote angles: α (alpha), β (beta), θ (theta) along with capital letters A, B, C • If 2 angles have the same initial side and the same terminal side, then they are coterminal • Picture 6.4 (look at both pictures) Measurement of an angle Comes from the rotation and how much of the circle it rotates around. Degree- most common measurement 1°= 1/360 P. 455 Types of Angles Acute angles Right Angles Obtuse angles Straight angles 0 up to 90 90 90 up to 180 180 “θ lies in quadrant” • This is the abbreviation for what quadrant the terminal side of an angle is in when the angle is in standard position What quadrant does 0°, 90°, 180°, and 270° lie in? • They are not in a quadrant because they are on the axis To find an angle coterminal to a given angle add or subtract 360°. • 30° – 30°(+360) coterminal to 390° • 30° – 30°(+720) coterminal to 750° • 30° – 30°(+n(360)) will be coterminal when n is an integer • Complementary – 2 angles that total to 90° • Supplementary – 2 angles that total to 180° You must use positive angles for these! Look at Ex. 2 p. 456 Parts of a degree • Fractional parts of degrees are historically denoted in minutes ( )׳and seconds (˝) • You can use the calculator to change these parts to decimal degrees • 1´ = (1/60) 1° • 1˝ = (1/3600) 1° EXAMPLE • 64 degrees 32 minutes 47 seconds • 64°32´47˝ • To enter into your calculator 64 2nd key, then apps (angle), then °, enter 32 2nd key, then apps, then ´, then enter 47 then alpha key, then + key (˝), then enter Radian Measure • Use in pre-cal. • Comes from the central angle of a circle – Central angle – angle whose vertex is at the center of a circle Radian • Measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle P. 457 6.12 1 full revelolution = 2π • Since s and r have the same units it is a ratio. So unitless 90° is equivalent to π/2 180° is equivalent to π To find complement and supplement of an angle • Complements totaled to 90° – So in degrees to find the complement of an angle we subtract from 90. – In radians we subtract from π/2 since it is the equivalent of 90 • Supplements total to 180° – In degrees to find the supplement we subtract from 180. – So in radians we subtract from π since it is equivalent to 180 Coterminal • In degrees it is found by adding or subtracting 360 (or a multiple of 360) • In radians it is found by adding or subtracting 2π (or a multiple of 2π) since it is the equivalent of 360