Angles and Their Measures 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 0 up to 90 Right Angles 90 Obtuse angles 90 up to 180 Straight angles 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°(+720) coterminal to 750° 30°(+n(360)) will be coterminal when n is an integer

Complementary Supplementary 2 angles that total to 90° 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