Problem 33 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(-\frac{5 \pi}{4}\)

Step-by-Step Solution

Answer

Positive: \(\frac{3\pi}{4}\), Negative: \(-\frac{13\pi}{4}\)

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides when drawn in standard position. To find coterminal angles, you add or subtract multiples of a complete rotation (in radians, multiples of \(2\pi\)).

Step 2: Finding a Positive Coterminal Angle

To find a positive coterminal angle with \(-\frac{5\pi}{4}\), we add \(2\pi\) to it.\(-\frac{5\pi}{4} + 2\pi = -\frac{5\pi}{4} + \frac{8\pi}{4} = \frac{3\pi}{4}\).So, \(\frac{3\pi}{4}\) is a positive coterminal angle.

Step 3: Finding a Negative Coterminal Angle

To find another negative coterminal angle, subtract \(2\pi\) from \(-\frac{5\pi}{4}\).\(-\frac{5\pi}{4} - 2\pi = -\frac{5\pi}{4} - \frac{8\pi}{4} = -\frac{13\pi}{4}\).Thus, \(-\frac{13\pi}{4}\) is another negative coterminal angle.

Key Concepts

Angle Measurement

In trigonometry, angle measurement is a crucial concept needed to understand and work with angles. Angles can be measured in different units, primarily in degrees and radians. Knowing how to convert between these units is key:

• Degrees are a measure of angle based on dividing a circle into 360 equal parts.
• Radians are another angle measurement, where one full circle equates to \(2\pi\) radians.

When dealing with angles, it's important to consider whether they are positive or negative. A positive angle represents a counter-clockwise rotation from the initial side, while a negative angle indicates a clockwise rotation. Additionally, any angle can have angles that are coterminal with it. This means they share the same initial and terminal sides, even if their measurements appear different.

Radians

Radians are a unit of angular measure used extensively in mathematics, especially in calculus and trigonometry. A radian describes the angle subtended by an arc that is equal in length to the radius of the circle. Here’s what you need to know about radians:

• There are \(2\pi\) radians in a full circle, meaning \(360\) degrees is equivalent to \(2\pi\) radians.
• To convert from degrees to radians, you can use the formula: \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \).
• Conversely, to convert radians to degrees: \( \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} \).

Understanding radians is vital for applying trigonometric functions, particularly when solving problems involving rotational dynamics and wave functions. Radians simplify many mathematical equations and are preferred in advanced mathematics settings.

Trigonometry

Trigonometry is the branch of mathematics that focuses on the relationships between the angles and lengths of triangles. It is incredibly useful for solving real-world problems and understanding the geometry and behavior of waves and oscillations. Key trigonometric notions include:

• Trigonometric functions like sine, cosine, and tangent, which relate the angles and sides of a right-angled triangle.
• Coterminal angles, which occur when two angles have equivalent terminal points even if their measurements are different. They are found by adding or subtracting full rotations (like \(2\pi\) radians).
• Understanding the unit circle, which is a circle of radius one centered at the origin of the coordinate plane. It helps in visualizing angle measures in radians and aids in calculating trigonometric functions.

By mastering trigonometry, you can solve problems across various fields of science and engineering. It's crucial for analyzing periodic phenomena, such as waves, and for understanding rotations in geometry and physics.