Verified Answer :

  • Simplify \(\frac{15\pi}{4}\) to the coterminal angle \(\frac{7\pi}{4}\), yielding \(\sin(\frac{15\pi}{4}) = -\frac{\sqrt{2}}{2}\), \(\cos(\frac{15\pi}{4}) = \frac{\sqrt{2}}{2}\), and \(\tan(\frac{15\pi}{4}) = -1\).
  • Identify \(120^\circ\) as a second-quadrant angle with a reference angle of \(60^\circ\).
  • Calculate \(\sin(120^\circ) = \frac{\sqrt{3}}{2}\).
  • Final results: \(\sin(\frac{15\pi}{4}) = \boxed{-\frac{\sqrt{2}}{2}}\), \(\cos(\frac{15\pi}{4}) = \boxed{\frac{\sqrt{2}}{2}}\), \(\tan(\frac{15\pi}{4}) = \boxed{-1}\), \(\sin(120^\circ) = \boxed{\frac{\sqrt{3}}{2}}\).

Explanation

  1. Problem Analysis

We are tasked with finding the exact values of trigonometric functions for two given angles. The first angle is \(\theta_1 = \frac{15\pi}{4}\), for which we will determine the sine, cosine, and tangent values. The second angle is \(\theta_2 = 120^\circ\), for which we will calculate the sine value. We will use the unit circle properties, reference angles, and quadrant signs (ASTC) to evaluate these precisely.

  1. Evaluating Trigonometric Functions for \(\frac{15\pi}{4}\)

To evaluate the trigonometric functions for \(\theta_1 = \frac{15\pi}{4}\), we first simplify the angle by finding its coterminal angle within the interval \([0, 2\pi)\). Subtracting \(2\pi\) (or \(\frac{8\pi}{4}\)):
\(\frac{15\pi}{4} - 2\pi = \frac{15\pi}{4} - \frac{8\pi}{4} = \frac{7\pi}{4}\).
The angle \(\frac{7\pi}{4}\) lies in the fourth quadrant. The reference angle is \(2\pi - \frac{7\pi}{4} = \frac{\pi}{4}\).
In the fourth quadrant, the cosine is positive, while the sine and tangent are negative. Thus:
\(\sin\left(\frac{15\pi}{4}\right) = \sin\left(\frac{7\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)
\(\cos\left(\frac{15\pi}{4}\right) = \cos\left(\frac{7\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)
\(\tan\left(\frac{15\pi}{4}\right) = \tan\left(\frac{7\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) = -1\)

  1. Evaluating \(\sin 120^{\circ}\)

To evaluate \(\sin(120^{\circ})\), we note that \(120^{\circ}\) is located in the second quadrant. The reference angle is \(180^{\circ} - 120^{\circ} = 60^{\circ}\). Since sine is positive in the second quadrant, we have:
\(\sin(120^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2}\)

  1. Conclusion

The trigonometric values have been determined:
For \(\frac{15\pi}{4}\): \(\sin\left(\frac{15\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), \(\cos\left(\frac{15\pi}{4}\right) = \frac{\sqrt{2}}{2}\), and \(\tan\left(\frac{15\pi}{4}\right) = -1\).
For \(120^{\circ}\): \(\sin(120^{\circ}) = \frac{\sqrt{3}}{2}\).

Examples

Trigonometric functions are essential in fields like engineering and physics for modeling periodic phenomena. For example, when calculating the voltage in an AC circuit at a specific phase angle or determining the height of a pendulum at a given time, these exact values are critical. By converting complex angles into their reference angles, you can quickly determine the state of a system without needing a calculator, ensuring precision in design and analysis.

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