Lesson 5-1 Angles and Degree Measure Objective: To convert decimal degrees to measure degrees. To find the number of degrees in a given number of rotations. Identify coterminal angles.

Trigonometry The branch of mathematics that studies triangles. It deals with the relationship between the sides and the angles. In order to do this you must also understand the relationship between angles and circles.

Angles An angle is formed by two rays with a common endpoint. (geometry definition) An angle is the result of a rotation of a ray about its endpoint. (trigonometry definition)

Standard Position When the initial side is on the positive x axis and the endpoint is on the origin Then the angle is in standard position.

Positive Angle – Standard Form α

Negative Angle – Standard Form β

Quadrant I Where the terminal side lays is where the angle is said to lie. Between 0 o and 90 o or 0 and

Quadrant II Between 90 o and 180 o or and

Quadrant III Between 180 o and 270 o or and

Quadrant IV Between 270 o and 360 o or and

Quadrantal When the terminal side lies on an axis it is called a quadrantal.

Angular Measurement Degree - of a complete rotation in the counterclockwise direction. 1o1o

Decimal-Degrees and Degree-Minute-Second Form There are 2 forms for expressing degrees: –decimal degrees –degree-minute-second

Basic Conversions: 1 degree= 60 minutes 1 minute = 60 seconds 1 degree=3600 seconds 1 o = 60’ 1’=60” 1 o = 3600’’

Example Convert 0.5 o to Minutes 0.5 o 0.5 Degrees 1 degree=60 minutes 30 minutes

Example: Convert o to Degree-Minute- Second degrees 1 degree=60 min. 38 degrees minutes 1 min. = 60 sec.

Example: Convert 30’ to Decimal Degrees 1 minute = 1/60 degree

Example: Convert 28 o 32’ 45” to Decimal Degrees 28 degrees, 32 minutes, 45 seconds 1 minute=1/60 degree 1 second=1/3600 degree

Coterminal Angles Coterminal – two angles that have a terminal side that is in the same position. (ex: +30 o and -330 o or 60 o and 420 o )

Coterminal Angles Angle θ is coterminal with every angle: θ o n n is an integer that tells how many times around the circle.

Examples Find the positive angle that is coterminal with each of the following: 430 o 70 o 25 o 385 o

Reference Angle The reference angle θ ʹ (theta prime) for the angle θ is the acute angle that is formed by the terminal side of the angle and the x axis. (an acute angle means that it must be between 0 o and 90 o. θ θ θʹθʹ θʹθʹ

Reference Angle If is an acute angle and lies in Quadrant I then the reference angle is the same as θ. (θ ʹ = θ) If θ lies in Quadrant II then θ + θ ʹ = 180 o. If θ lies in Quadrant III then θ – θ ʹ = 180 o. If θ lies in Quadrant IV then θ + θ ʹ = 360 o. θ

Reference Angles Find the reference angle if θ is: 160 o 30 o -95 o 125 o