Standard Position, Coterminal and Reference Angles

Measure of an Angle 1 The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. -1 1 Initial Side Terminal Side -1

Angles that share the same initial and terminal sides. Coterminal Angles 1 Angles that share the same initial and terminal sides. Example: 30° and 390° -1 1 -1

Angles  and  are coterminal. Coterminal Angles Angles that have the same initial and terminal sides are coterminal. Angles  and  are coterminal.

Example of Finding Coterminal Angles You can find an angle that is coterminal to a given angle  by adding or subtracting multiples of 360º. Ex 1: Find one positive and one negative angle that are coterminal to 112º. For a positive coterminal angle, add 360º : 112º + 360º = 472º For a negative coterminal angle, subtract 360º: 112º - 360º = -248º

Ex 2. Find one positive and one negative angle that is coterminal with the angle  = 30° in standard position. Ex 3. Find one positive and one negative angle that is coterminal with the angle  = 272 in standard position.

Ex 4. Find one positive and one negative angle that is coterminal with the angle  = in standard position. Ex 5. Find one positive and one negative angle that is coterminal with the angle  = in standard position.

Reference Angles The values of the trigonometric functions of angles greater than 90 (or less than 0) can be determined from their values at corresponding acute angles called reference angles. 𝜃’ 𝜃

Reference Angles The reference angles for  in Quadrants II, III, and IV are shown below.  ′ =  –  (radians)  ′ = 180 –  (degrees)  ′ =  –  (radians)  ′ =  – 180 (degrees)  ′ = 2 –  (radians)  ′ = 360 –  (degrees)

Special Angles – Reference Angles

Example – Finding Reference Angles Find the reference angle  ′. a.  = 300 b.  = 2.3 c.  = –135

Example (a) – Solution Because 300 lies in Quadrant IV, the angle it makes with the x-axis is  ′ = 360 – 300 = 60. The figure shows the angle  = 300 and its reference angle  ′ = 60. Degrees

Example (b) – Solution cont’d Because 2.3 lies between  /2  1.5708 and   3.1416, it follows that it is in Quadrant II and its reference angle is  ′ =  – 2.3  0.8416. The figure shows the angle  = 2.3 and its reference angle  ′ =  – 2.3. Radians

Example (c) – Solution cont’d First, determine that –135 is coterminal with 225, which lies in Quadrant III. So, the reference angle is  ′ = 225 – 180 = 45. The figure shows the angle  = –135 and its reference angle  ′ = 45. Degrees

Reference Angles When your angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0° and 360° or 0 and 2𝜋.

Your Turn: Find the reference angle for each of the following. 213° 1.7 −144° -144 ̊ is coterminal to 216 ̊ 216 ̊ - 180 ̊ = 36 ̊