Answered

Identify the reference angle $\phi$ for each given angle, $\theta$.



When $\theta=300^{\circ}, \phi=$ □ degrees.



When $\theta=225^{\circ}, \phi=$ □ degrees.



When $\theta=480^{\circ}, \phi=$ □ degrees.



When $\theta=-210^{\circ}, \phi=$ □ degrees.

Verified Answer :

  • Determine the coterminal angle \(\theta'\) for each given angle \(\theta\) such that \(0^{\circ} \le \theta' < 360^{\circ}\).
  • Identify the quadrant of the terminal side of \(\theta'\).
  • Calculate the reference angle \(\phi\) as the acute angle formed between the terminal side and the nearest x-axis.
  • Final values: \(\theta=300^{\circ} \to \phi=60^{\circ}\), \(\theta=225^{\circ} \to \phi=45^{\circ}\), \(\theta=480^{\circ} \to \phi=60^{\circ}\), \(\theta=-210^{\circ} \to \phi=30^{\circ}\).
  • Final answers: \(\boxed{60^{\circ}}\), \(\boxed{45^{\circ}}\), \(\boxed{60^{\circ}}\), \(\boxed{30^{\circ}}\).

Explanation

  1. Definitions and methodology for finding reference angles.

The reference angle \(\phi\) for an angle \(\theta\) in standard position is defined as the smallest positive acute angle that the terminal side of \(\theta\) makes with the x-axis. This angle always falls in the range \([0^{\circ}, 90^{\circ}]\). To find the reference angle for any given angle, we first determine its coterminal angle \(\theta'\) such that \(0^{\circ} \le \theta' < 360^{\circ}\). Then, we use the quadrant in which \(\theta'\) lies to calculate \(\phi\):

  • If \(\theta'\) is in Quadrant I, \(\phi = \theta'\).
  • If \(\theta'\) is in Quadrant II, \(\phi = 180^{\circ} - \theta'\).
  • If \(\theta'\) is in Quadrant III, \(\phi = \theta' - 180^{\circ}\).
  • If \(\theta'\) is in Quadrant IV, \(\phi = 360^{\circ} - \theta'\).
  1. Calculating reference angles for each given \(\theta\).
  • For \(\theta_1 = 300^{\circ}\): This angle lies in Quadrant IV because \(270^{\circ} < 300^{\circ} < 360^{\circ}\). The reference angle is \(\phi_1 = 360^{\circ} - 300^{\circ} = 60^{\circ}\).
  • For \(\theta_2 = 225^{\circ}\): This angle lies in Quadrant III because \(180^{\circ} < 225^{\circ} < 270^{\circ}\). The reference angle is \(\phi_2 = 225^{\circ} - 180^{\circ} = 45^{\circ}\).
  • For \(\theta_3 = 480^{\circ}\): First, find the coterminal angle within \([0^{\circ}, 360^{\circ})\) by subtracting \(360^{\circ}\): \(480^{\circ} - 360^{\circ} = 120^{\circ}\). Since \(90^{\circ} < 120^{\circ} < 180^{\circ}\), it is in Quadrant II. The reference angle is \(\phi_3 = 180^{\circ} - 120^{\circ} = 60^{\circ}\).
  • For \(\theta_4 = -210^{\circ}\): Find the coterminal angle by adding \(360^{\circ}\): \(-210^{\circ} + 360^{\circ} = 150^{\circ}\). Since \(90^{\circ} < 150^{\circ} < 180^{\circ}\), it is in Quadrant II. The reference angle is \(\phi_4 = 180^{\circ} - 150^{\circ} = 30^{\circ}\).
  1. Final values for reference angles.

The reference angles for the given values are:

  • When \(\theta = 300^{\circ}\), \(\phi = 60^{\circ}\).
  • When \(\theta = 225^{\circ}\), \(\phi = 45^{\circ}\).
  • When \(\theta = 480^{\circ}\), \(\phi = 60^{\circ}\).
  • When \(\theta = -210^{\circ}\), \(\phi = 30^{\circ}\).

The final answers are \(\boxed{60^{\circ}}\), \(\boxed{45^{\circ}}\), \(\boxed{60^{\circ}}\), and \(\boxed{30^{\circ}}\).

Examples

Reference angles are incredibly useful in trigonometry for evaluating trigonometric functions of angles larger than \(90^{\circ}\). For example, if you are calculating the height of a swing set at various angles, knowing that the reference angle for \(300^{\circ}\) is \(60^{\circ}\) allows you to determine that \(\sin(300^{\circ}) = -\sin(60^{\circ}) = -\frac{\sqrt{3}}{2}\). This simplifies complex physical problems into basic geometric calculations based on right triangles.

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