Evaluate. Write your answers in simplified, rationalized forms

Evaluate. Write your answers in simplified, rationalized forms. Do not round.
sin 30° =
COS 30°=
tan 30° =

The correct answer and explanation is:

The values for sin 30°, cos 30°, and tan 30° are commonly known, and they are derived from the properties of a 30°-60°-90° triangle, which has specific ratios for its sides.

sin 30°:

For a 30° angle in a right triangle, the opposite side is half the length of the hypotenuse. The ratio of the opposite side to the hypotenuse is defined as sine. In a 30°-60°-90° triangle, the sides have a ratio of 1:√3:2. Therefore, the sine of 30° is: sin⁡30°=12\sin 30° = \frac{1}{2}

cos 30°:

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. For a 30°-60°-90° triangle, the adjacent side for 30° is √3, and the hypotenuse is 2. So: cos⁡30°=32\cos 30° = \frac{\sqrt{3}}{2}

tan 30°:

The tangent of an angle is the ratio of the opposite side to the adjacent side. In a 30°-60°-90° triangle, for the 30° angle, the opposite side is 1, and the adjacent side is √3. Therefore: tan⁡30°=13\tan 30° = \frac{1}{\sqrt{3}}

To rationalize the denominator: tan⁡30°=13×33=33\tan 30° = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}

Summary of answers:

• sin⁡30°=12\sin 30° = \frac{1}{2}
• cos⁡30°=32\cos 30° = \frac{\sqrt{3}}{2}
• tan⁡30°=33\tan 30° = \frac{\sqrt{3}}{3}

These values are derived from the special properties of a 30°-60°-90° triangle, where the ratios of the sides are consistent and can be used to determine trigonometric values for these angles.