Questions: Trigonometric Functions Trigonometric functions and special angles: Problem type 1: Degrees Find the exact value of sin 120°. sin 120°= √

Solution Steps

To find the exact value of \(\sin 120^\circ\), we can use the fact that \(120^\circ\) is in the second quadrant where the sine function is positive. We can express \(120^\circ\) as \(180^\circ - 60^\circ\). Using the sine subtraction identity, \(\sin(180^\circ - \theta) = \sin \theta\), we find that \(\sin 120^\circ = \sin 60^\circ\). The exact value of \(\sin 60^\circ\) is \(\frac{\sqrt{3}}{2}\).

To find the sine of \(120^\circ\), we first convert the angle from degrees to radians. The conversion formula is: \[ \text{radians} = \frac{\pi}{180} \times \text{degrees} \] For \(120^\circ\), this gives: \[ \text{radians} = \frac{\pi}{180} \times 120 = \frac{2\pi}{3} \approx 2.0944 \]

The sine of an angle in radians can be calculated using the sine function. For \(\frac{2\pi}{3}\), the sine value is approximately: \[ \sin\left(\frac{2\pi}{3}\right) \approx 0.8660 \]

Since \(120^\circ\) is in the second quadrant, where the sine function is positive, we can use the identity \(\sin(180^\circ - \theta) = \sin \theta\). Thus: \[ \sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2} \] The exact value of \(\sin 60^\circ\) is \(\frac{\sqrt{3}}{2} \approx 0.8660\).

Final Answer