Problem 62 - Algebra and Trigonometry

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$ -760^{\circ} $$

Step-by-Step Solution

Answer

The positive coterminal angle less than \(360^{\circ}\) that is coterminal with \(-760^{\circ}\) is \(320^{\circ}\)

Step 1: Understand the problem

We want to find an equivalent angle to \(-760^{\circ}\) which falls within the usual scale for an angle in trigonometry: \(0^{\circ} - 360^{\circ}\) or \(0 - 2\pi\) in radians.

Step 2: Add \(360^{\circ}\) to the given angle

First, add \(-760^{\circ} + 360^{\circ}\) to obtain \(-400^{\circ}\)

Step 3: Continue adding \(360^{\circ}\)

Then, add \(-400^{\circ} + 360^{\circ}\) to obtain \(-40^{\circ}\)

Step 4: Final addition

Finally, add \(-40^{\circ} + 360^{\circ}\) to get \(320^{\circ}\)

Key Concepts

Positive Angles in Trigonometry

Angles are fundamental in the study of trigonometry and can be depicted in various ways, depending on their size and direction. A positive angle in trigonometry is an angle measured by a counterclockwise rotation from the positive x-axis. The concept of positive angles is crucial because it sets a standard direction for measuring angles, which is essential for consistency in trigonometric calculations.

When working with positive angles, it's important to remember that they continue beyond one full rotation, which is equal to either a full circle of 360 degrees or \(2\pi\) radians. This means that an angle of 370 degrees is coterminal with 10 degrees because they share the same terminal side. Coterminal angles, such as this example, may be positive or negative, but can always be adjusted to fall within the primary interval from 0 to 360 degrees or 0 to \(2\pi\) radians.

Angle Measurement Conversions

Understanding angle measurement conversions is key to solving problems in trigonometry. There are two primary units of measurement for angles: degrees and radians. Converting between these units is a frequently used skill. One complete rotation around a circle is 360 degrees, which is equivalent to \(2\pi\) radians. Thus, the conversion factor between degrees and radians is \(\frac{\pi}{180}\).

For example, to convert 180 degrees to radians, you would calculate \(180^\circ \times \frac{\pi}{180} = \pi\) radians. Conversely, to convert \(\pi\) radians to degrees, you would use the formula \(\pi \times \frac{180}{\pi} = 180^\circ\). This conversion is essential when dealing with trigonometric functions, which can have different representations or results depending on the unit of measure.

Trigonometric Functions

Trigonometric functions are a core component of trigonometry and are used to relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which are based on right-angled triangles. There are also reciprocal functions: cosecant (csc), secant (sec), and cotangent (cot), which are the reciprocals of sine, cosine, and tangent, respectively.

These functions are periodic and they repeat their values in cycles of 360 degrees or \(2\pi\) radians. This is why understanding coterminal angles is so significant, as trigonometric functions will have the same value for coterminal angles. For instance, the sine of 30 degrees and the sine of 390 degrees (which is 30 degrees plus one full rotation) are the same. Mastery of these functions and their properties is essential to solving more complex problems in trigonometry.