Trigonometry Using Co-terminal and Reference Angles Dr. Shildneck Fall, 2014

Co-Terminal Angles An angle is co-terminal to another angle if their terminal sides are end at the same position. The trigonometric ratios regarding co-terminal angles are equivalent.

Co-Terminal Angles o

Co-Terminal Angles - Degrees 60 0 – 360 = π = 420 o – 360 – 360 =

Co-Terminal Angles - Radians π/3 – 2 π = -5 π/3 + 2 π = 7 π/3 – 2 π – 2 π = -11 π/3π/3

Reference Angles The reason that the trigonometry for co-terminal angles are the same is because they have the same reference angles. Reference angles allow us to utilize right- triangle trigonometry on a rotational system

Reference Angles A reference angle is the angle that the terminal side makes with the “closest” part of the x-axis. Reference angles always have a measure between zero and 90 degrees (0 and π/2 radians). Note: Always form the right angle of a triangle by dropping a perpendicular to the x-axis. (The right angle always touches the x-axis.)

Reference Angles (in degrees) 60 o 120 o -135 o 690 o 60 o 45 o 30 o

Reference Angles (in radians) π/3 2 π/3 5 π/4 23 π/6 π/3 π/4 π/6 0 π, 2π, 4π, … π

Reference Angles To determine a reference angle for an angle in standard position, compare the angle’s measure to the closest x-axis. You will either need to subtract using a multiple of 180 ( π) or multiple of 360 (2 π). Remember, reference angles always have a measure between zero and 90 degrees (0 and π/2 radians).

ASSIGNMENT Page 238 #18-26 Page 251 #17-24