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Chapter 6: Problem 19

Find the exact value of the trigonometric function. $$\tan 750^{\circ}$$

Short Answer

Expert verified
\( \tan 750^{\circ} = \frac{1}{\sqrt{3}} \).

Step by step solution

01

Understand the Angle

The angle given is 750 degrees, which is larger than 360 degrees, meaning it's more than one full rotation around the circle.
02

Find the Coterminal Angle

To simplify the angle, we find a coterminal angle that is within the 0 to 360-degree range by subtracting 360 degrees. \[750^{\circ} - 360^{\circ} = 390^{\circ}\]However, 390 degrees is still greater than 360 degrees, so we subtract another 360 degrees:\[390^{\circ} - 360^{\circ} = 30^{\circ}\]Therefore, 750 degrees is coterminal with 30 degrees.
03

Evaluate the Trigonometric Function

Now that we have the coterminal angle, 30 degrees, we find \( \tan 30^{\circ} \). The value is obtained from the known trigonometric values:\[\tan 30^{\circ} = \frac{1}{\sqrt{3}}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coterminal Angles
Coterminal angles are angles that share the same terminal side after spin or rotation. Essentially, they are angles that differ by a full circle, which is 360 degrees. To find a coterminal angle, particularly with bigger angles like 750 degrees, you simply need to keep subtracting or adding 360 degrees until you bring the angle into the standard range of 0 to 360 degrees.For example:
  • Subtract 360 degrees from 750 degrees: \[750^{\circ} - 360^{\circ} = 390^{\circ}\]
  • 390 degrees is still greater than 360, so we subtract again:\[390^{\circ} - 360^{\circ} = 30^{\circ}\]
This process makes it manageable to find other trigonometric functions without needing to think about large rotations.
Tangent Function
The tangent function is one of the primary trigonometric functions, alongside sine and cosine. It is particularly important in right-angled triangles and is defined as the ratio of the opposite side to the adjacent side.This means:\[\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} \]In terms of a unit circle, where the hypotenuse is the radius, the tangent of an angle is the length of the segment of the tangent line that subtends the angle from the point.Specifically, for \(30^{\circ}\), we use the special angles' known values:
  • \(\tan 30^{\circ} = \frac{1}{\sqrt{3}}\)
It's a precise value that comes in handy often and is good to remember, especially for trigonometry exams.
Angle Measurement
Understanding angle measurement is crucial when working with trigonometric functions. Angles can be measured in degrees or radians, but degrees are more commonly used in basic trigonometry problems. A full circle is 360 degrees, which marks a complete rotation. The significance of 360 is its divisibility by many numbers, which historically made it useful for various calculations. Here are some useful facts about angles:
  • Quarter circle (right angle) = 90 degrees
  • Straight angle = 180 degrees
  • Full rotation = 360 degrees
Working with angles beyond 360 degrees involves finding coterminal angles, as these angles help simplify the computation by linking excessive rotations back into the standard circle measurement range.

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