a) Could angle Theta be defined as the reference angle for an angle in standard position? b) If Theta =20^circ ,determine the measures of the angles formed by the arm of the pendulum as an angle in standard position. c) What would the measure of the reference angle be for the angles in standard position given Theta =20^circ

**a) Can θ be a reference angle?**<br /><br />Yes, θ can be defined as the reference angle for an angle in standard position. A reference angle is the acute angle formed between the terminal side of an angle in standard position and the x-axis. Since θ is shown as an acute angle measured from the positive x-axis, it fits the definition of a reference angle.<br /><br />**b) Measures of angles in standard position given θ = 20°**<br /><br />Since the pendulum swings symmetrically about the vertical axis, the angles in standard position formed by the arm are:<br /><br />* **20°:** This is when the pendulum is 20° counterclockwise from the positive x-axis.<br />* **180° - 20° = 160°:** This is when the pendulum has swung to the other side, which is 20° counterclockwise from the negative x-axis (or 160° counterclockwise from the positive x-axis).<br />* **180° + 20° = 200°:** This is when the pendulum has swung back to the original side, but past the vertical, 20° clockwise from the negative x-axis.<br />* **360° - 20° = 340°:** This is when the pendulum is 20° clockwise from the positive x-axis.<br /><br />These four angles represent the pendulum's position in each of the four quadrants. Further swings would result in coterminal angles (angles with the same terminal side).<br /><br />**c) Reference angle for angles in standard position given θ = 20°**<br /><br />The reference angle is always the acute angle between the terminal side of the angle and the x-axis. Since we're given that θ = 20°, and we've determined the standard position angles are 20°, 160°, 200°, and 340°, the reference angle for *all* of these is 20°.<br />