Understanding the Exact Value of Tan 150 Degrees

In trigonometry, the tangent function (tan) is a fundamental concept that helps us understand the relationships between angles and sides in right triangles and their applications in various fields such as physics, engineering, and architecture. One specific case that is often asked is the exact value of tan 150 degrees. In this article, we will explore how to find the exact value of tan?150°.

Using the Unit Circle and Quadrant Information

Tan 150 degrees can be determined using the properties of the unit circle and quadrant information. When an angle is in the second quadrant (between 90° and 180°), the tangent function is negative because the sine value is positive and the cosine value is negative.

First, we can express 150 degrees as:

150°180°-30°

Therefore, we can use the tangent subtraction formula:

-tan30°

Using the Subtraction Formula and Known Values

We know that:

tan30°13

Thus, we can substitute this value into the equation:

-tan30°-13-33

Therefore, the exact value of tan?150° is:

boxed{-33}

Using the Unit Circle Trigonometric Values

Another way to find the exact value of tan 150 degrees is by using the unit circle and specific trigonometric values. Recall that the sine and cosine values of 30 degrees on the unit circle are:

sin150°12 and cos150°-32

Therefore, we can write:

tan150°sin150cos15012-32

Simplifying this expression results in:

tan150°-13-33

Key Concepts and Mnemonics

To remember the signs of trigonometric functions in different quadrants, we use the ASTC mnemonic:

A All (sine, cosine, tangent are positive in the first quadrant) S Sine (sine is positive in the second quadrant) T Tangent (tangent is positive in the third quadrant) C Cosine (cosine is positive in the fourth quadrant)

Additionally, we can use the angle transformation 150°180°-30°. By using trigonometric identities and angle properties, we can determine the exact value of tangent for any angle.

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