What are the sine, cosine, and tangent values of \( 30^{\circ} \)?

The sine, cosine, and tangent of a \(30^{\circ}\) angle can be determined using the properties of a 30-60-90 right triangle or the unit circle. Below are the exact values along with their decimal approximations: ### Exact Values: - **Sine (\(\sin 30^{\circ}\))** \[ \sin 30^{\circ} = \frac{1}{2} \] - **Cosine (\(\cos 30^{\circ}\))** \[ \cos 30^{\circ} = \frac{\sqrt{3}}{2} \] - **Tangent (\(\tan 30^{\circ}\))** \[ \tan 30^{\circ} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \] ### Decimal Approximations: - **\(\sin 30^{\circ} \approx 0.5\)** - **\(\cos 30^{\circ} \approx 0.8660\)** - **\(\tan 30^{\circ} \approx 0.5774\)** ### Explanation: Consider a 30-60-90 right triangle, where the sides are in the ratio \(1 : \sqrt{3} : 2\): - **Opposite Side to \(30^{\circ}\):** 1 - **Adjacent Side to \(30^{\circ}\):** \(\sqrt{3}\) - **Hypotenuse:** 2 Using these ratios: - **\(\sin 30^{\circ}\)** is the ratio of the opposite side to the hypotenuse: \[ \sin 30^{\circ} = \frac{1}{2} \] - **\(\cos 30^{\circ}\)** is the ratio of the adjacent side to the hypotenuse: \[ \cos 30^{\circ} = \frac{\sqrt{3}}{2} \] - **\(\tan 30^{\circ}\)** is the ratio of the opposite side to the adjacent side: \[ \tan 30^{\circ} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \] Understanding these values is fundamental in trigonometry and has various applications in fields such as physics, engineering, and architecture.