Problem 14 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Rewrite each of the following expressions in terms of a positive acute angle. This positive acute angle is sometimes referred to as a reference angle. (a) \(\tan 200^{\circ}\) (b) \(\tan \left(-200^{\circ}\right)\)

Step-by-Step Solution

Answer

The equivalent positive acute angles (reference angles) for \( \tan 200^{\circ} \) and \( \tan \left(-200^{\circ}\right) \) are both \( 160^{\circ} \).

Step 1: Find Reference Angle for \( \tan 200^{\circ} \)

To find the reference angle for \( \tan 200^{\circ} \), note that this angle is in the third quadrant where tangent is positive. Subtract the actual angle from 360° to find an equivalent angle that's in the first quadrant: \( 360^{\circ} - 200^{\circ} = 160^{\circ} \).

Step 2: Find Reference Angle for \( \tan \left(-200^{\circ}\right) \)

To find the reference angle for \( \tan \left(-200^{\circ}\right) \), note that a negative angle goes in the opposite direction on the unit circle. Therefore, we add the value of the angle to 360° to find the equivalent positive angle: \( 360^{\circ} + \left(-200^{\circ}\right) = 160^{\circ} \). The reference angle in the first quadrant is the same as the previous step.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. It's particularly concerned with right-angled triangles where one angle is 90 degrees. The primary functions in trigonometry are sine, cosine, and tangent, often abbreviated as sin, cos, and tan respectively. Each of these functions is a ratio of two sides of a triangle and relates to an angle of the triangle.

Understanding trigonometry is essential because it's used not only in mathematics but also in various fields such as physics, engineering, and geography. For instance, engineers use trigonometry to calculate forces, while architects might use it to design buildings. One key application in trigonometry is determining the value of an angle or the length of a side in a triangle when some other aspect of the triangle is known.

Tangent Function

The tangent function, often referred to by its abbreviation tan, is one of the primary trigonometric functions. For an angle formed by a right-angled triangle, the tangent of that angle is the ratio of the length of the opposite side to the length of the adjacent side. This can be remembered by the mnemonic 'TOA' where Tangent equals Opposite over Adjacent.

The tangent function is periodic and has a period of 180 degrees, or \(\pi\) radians. It is undefined for angles where the cosine is zero (for example, 90 degrees or \(\frac{\pi}{2}\) radians) because the adjacent side in the right triangle would have a length of zero, resulting in a division by zero. In terms of the unit circle, tangent is also the y-coordinate divided by the x-coordinate of a point on the circle's circumference, which corresponds to a given angle. Understanding how the tangent function behaves is particularly important when dealing with angles that exceed 90 degrees because the values can be positive or negative depending on the angle's quadrant.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. The unit circle allows us to define the trigonometric functions geometrically, which is particularly useful when dealing with angles beyond the standard first quadrant (0 to 90 degrees).

Quadrants and Signs

There are four quadrants in the coordinate plane, each representing a range of angles with specific sign rules for sine, cosine, and tangent. For instance, the tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants.

Reference Angles

A reference angle is the acute angle formed by the terminal side of an angle in standard position and the x-axis. Every angle in standard position (counterclockwise from the positive x-axis) has a reference angle, which is always between 0 and 90 degrees or 0 and \(\frac{\pi}{2}\) radians. Reference angles are extremely helpful in determining the values of trigonometric functions for angles of any size since the values for sine, cosine, and tangent for an angle are the same as for its reference angle, except possibly for a sign change.