5.1 Angles and Degree Measures

Definitions An angle is formed by rotating one of two rays that share a fixed endpoint know as the vertex. The initial side of the angle is the ray that is fixed. The terminal side of the angle is the ray that rotates. Terminal Side Initial Side Vertex

The measure of an angle tells: 1. The direction of the rotation. 2. The amount of rotation to move from the initial side to the terminal side. Counter-clockwise rotation makes a positive angle Clockwise rotation makes a negative angle.

An angle with its vertex at the origin and its initial side along the positive x axis is said to be in standard position. Angles are measured in degrees and radians

You MUST be familiar with measures between 0° and 360°

Positive angles

Negative Angles

Examples: Draw an angle of 150° in standard position. Draw an angle of 225° in standard position. Draw an angle of -30° in standard position. Draw an angle of - 300° in standard position.

A full rotation around a circle is 360°. Measures more than 360° represent multiple rotations. 2 “rotations ”

Give the angle measure represented by each rotation. 5.5 rotations clockwise -1980° 3.3 rotations counterclockwise 1188° 9.5 rotations clockwise -3420° 6.75 rotations counterclockwise. 2430°

A quadrantal angle is one where the terminal side is one of the axes.

Two angles in standard position are called coterminal angles if they have the same terminal side. Every angle has infinitely many coterminal angles. These can be found by adding or subtracting multiples of 360°

Find one positive and one negative angle that is coterminal with: 45° 225° 135° 300°

Identify ALL angles that are coterminal with each angle 30° 120° 270° 330°

Determine a coterminal angle that is between 0° and 360°. State the quadrant in which the terminal side lies. 775° 55°; QI -1297° 143°; QII 595° 235°; QIII -777° 303°; QIV

A reference angle is an ACUTE angle formed by the terminal side of the given angle and the x axis.

Find the reference angle. 120° 60° -135° 45° 210° 30° 300° 60°

Exit Pass 1.Find one positive and one negative coterminal angle with 218°. 2.Find the reference angle for 218° 3.Draw a 300° angle in standard position. 4.Give 2 quadrantal angles.

Degrees, Minutes, Seconds The most common unit used to measure angles is the degree (°). The degree is subdivided into 60 equal parts known as minutes (‘). The minute is subdivided into 60 equal parts known as seconds (“). The concept of degree measurement comes from the Babylonian culture. Their number system was based off of 60 instead of 10 like we do today

Changing from Degrees to Degrees, Minutes, and Seconds (DMS) Change ° to degrees, minutes, and seconds. 15° + (0.735  60)’ 15° ’ 15° + 44’ + (0.1  60)” 15°+44’ + 6” 15°44’6”

Change from Degrees to DMS ° -16° 45’ ° 27° 27’ 54”

Change from DMS to Degrees Write 39° 5’ 34” as a decimal rounded to the nearest thousdandth. 39° + 5(1/60) + 34 (1/3600) °

Change from DMS to Degrees 23° 14’ 30” ° 223°25’15” °