Problem 30 - Precalculus : Building Concepts and Connections

Question

Find two angles that are coterminal with it. $$\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The two coterminal angles with \( \frac{3 \pi}{4} \) are \( \frac{11 \pi}{4} \) and \( -\frac{5 \pi}{4} \).

Step 1: Add a Full Rotation

Add \( 2\pi \) to \( \frac{3\pi}{4} \) to get a positive coterminal angle. The calculation becomes : \( \frac{3 \pi}{4} + 2\pi = \frac{3 \pi}{4} + \frac{8 \pi}{4} = \frac{11 \pi}{4} \)

Step 2: Subtract a Full Rotation

Subtract \( 2\pi \) from \( \frac{3\pi}{4} \) to get a negative coterminal angle. The calculation becomes : \( \frac{3 \pi}{4} - 2\pi = \frac{3 \pi}{4} - \frac{8 \pi}{4} = - \frac{5 \pi}{4} \)

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, specifically right-angled triangles. But its application goes beyond triangles, reaching into various fields like physics, engineering, astronomy, and even art. One of the crucial elements of trigonometry is understanding the concept of an angle and its measure. An angle in trigonometry is formed by the rotation from one ray to another, with the vertex being the point of rotation.

The measurement of angles can be in degrees or radians, two different systems used to express angular sizes. The angle provided in the exercise, \(\frac{3\pi}{4}\) radians, uses the radian measure, which is deeply tied to the concept of the circle. Here, the circumference of a unit circle—circle with a radius of one—is exactly \(2\pi\) radians. This plays a significant role when dealing with circular functions in trigonometry.

Circular Functions

Circular functions are trigonometric functions that are defined for all real numbers, unlike the basic trigonometric functions that are limited to angles between 0 and \(\pi/2\) radians. These functions include sine, cosine, tangent, cotangent, secant, and cosecant. They're called 'circular' because they can be represented as functions corresponding to points on a unit circle.

When dealing with problems involving circular functions, it's important to understand the concept of coterminal angles. Coterminal angles are angles that share the same initial and terminal sides but may differ by a full rotation, or multiple rotations, of \(2\pi\) radians or 360 degrees. In the given exercise, adding or subtracting \(2\pi\) radians from the angle \(\frac{3\pi}{4}\) results in angles that are coterminal with it, showing the periodic nature of circular functions.

Angular Measurements

Angular measurements are a way to quantify the size of an angle and they provide a systematic way to describe the orientation or rotation of a ray with respect to another ray. The most common units for measuring angles are degrees and radians.

In the exercise provided, adding or subtracting full rotations, in this case \(2\pi\) radians, to the initial angle of \(\frac{3\pi}{4}\), resulted in coterminal angles. This exercise emphasizes the importance of understanding that an angle measured in radians is equal to the length of the arc that the angle subtends on a unit circle. Understanding this relationship is critical in resolving problems that involve calculating the circumference, arc length, or areas related to circles. Coterminal angles highlight the periodicity of angles, meaning that adding any integer multiple of a full rotation doesnt alter the angular position, just as \(\frac{11\pi}{4}\) and \( -\frac{5\pi}{4}\) are coterminal with \(\frac{3\pi}{4}\).