Evaluate the following and indicate the reference angle 1

Evaluate the following and indicate the reference angle 1. Tan 330 degrees

The correct answer and explanation is:

Evaluate: tan⁡330∘\tan 330^\circ
Reference Angle: ?

Step 1: Find the reference angle

The reference angle is the acute angle formed between the given angle and the nearest x-axis. Since 330∘330^\circ is in the fourth quadrant (because it is between 270∘270^\circ and 360∘360^\circ), the reference angle is calculated by subtracting 330∘330^\circ from 360∘360^\circ: Reference angle=360∘−330∘=30∘\text{Reference angle} = 360^\circ – 330^\circ = 30^\circ

Step 2: Find tan⁡330∘\tan 330^\circ

The tangent function in the fourth quadrant is negative because tangent is positive in the first and third quadrants and negative in the second and fourth quadrants.

We use the reference angle of 30∘30^\circ to find the tangent value: tan⁡30∘=13=33\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}

Since 330∘330^\circ is in the fourth quadrant, tan⁡330∘\tan 330^\circ will be negative: tan⁡330∘=−tan⁡30∘=−33\tan 330^\circ = -\tan 30^\circ = -\frac{\sqrt{3}}{3}

Final answers:

tan⁡330∘=−33\tan 330^\circ = -\frac{\sqrt{3}}{3} Reference angle=30∘\text{Reference angle} = 30^\circ

Explanation

The tangent function is defined as the ratio of the sine to the cosine of the angle. Angles on the unit circle repeat every 360∘360^\circ, so angles greater than 360∘360^\circ or negative angles can be converted to equivalent angles between 0∘0^\circ and 360∘360^\circ.

The angle 330∘330^\circ lies in the fourth quadrant, where cosine is positive and sine is negative. Because tangent is sine divided by cosine, tangent in the fourth quadrant is negative.

To evaluate the tangent at 330∘330^\circ, the angle is related to the acute reference angle of 30∘30^\circ. Reference angles simplify trigonometric calculations by allowing the use of well-known values for acute angles.

Knowing tan⁡30∘=13\tan 30^\circ = \frac{1}{\sqrt{3}}, the sign of tangent at 330∘330^\circ is determined by the quadrant. Since the fourth quadrant tangent is negative, the answer becomes −13-\frac{1}{\sqrt{3}}, often rationalized to −33-\frac{\sqrt{3}}{3}.

The reference angle is essential for understanding the relationship between the given angle and its trigonometric function value without needing a calculator or a unit circle chart for every angle.