Problem 44 - Algebra and Trigonometry

Question

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 1270^{\circ} $$

Step-by-Step Solution

Answer

The angle coterminal with \(1270^{\circ}\) between \(0^{\circ}\) and \(360^{\circ}\) is \(190^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides, but may have different rotations. To find a coterminal angle within a specific range, like between \(0^{\circ}\) and \(360^{\circ}\), you can add or subtract multiples of \(360^{\circ}\) from the given angle.

Step 2: Set Up the Equation

We need to find \(1270^{\circ}\) minus a multiple of \(360^{\circ}\) to bring it within the \(0^{\circ}\) to \(360^{\circ}\) range. The formula we use will be: \[ 1270^{\circ} - k \times 360^{\circ} = \theta \] where \(\theta\) is the desired coterminal angle and \(k\) is an integer.

Step 3: Calculate the Multiple of 360

Find the integer \(k\) such that \(1270^{\circ} - k \times 360^{\circ}\) is between \(0\) and \(360\). This can be done by dividing \(1270\) by \(360\):\[ \frac{1270}{360} \approx 3.5278 \]The largest whole number less than or equal to \(3.5278\) is \(3\). So, \(k = 3\).

Step 4: Perform the Calculation

Substitute \(k = 3\) into the equation:\[ 1270^{\circ} - 3 \times 360^{\circ} \]Calculate:\[ 1270^{\circ} - 1080^{\circ} = 190^{\circ} \]So, \(190^{\circ}\) is a coterminal angle with \(1270^{\circ}\) within the range \(0^{\circ}\) to \(360^{\circ}\).

Step 5: Verify Result

Since \(190^{\circ}\) is between \(0^{\circ}\) and \(360^{\circ}\), it satisfies the condition needed for the coterminal angle. Therefore, the solution is verified and complete.

Key Concepts

Angle Measurement

Understanding angle measurement is an essential part of learning trigonometry. Angles are typically measured in degrees, which is a unit that divides a full circle into 360 equal parts. Each degree represents one of those parts of the circle. For instance, a right angle equals 90 degrees, which is a quarter of a full circle. You should remember that the circle is a complete rotation and is what defines the 360-degree measure.

• A complete circle is 360°.
• Half a circle, or a straight angle, is 180°.
• A quarter circle, or a right angle, is 90°.

These measurements help us understand positions and rotations relative to other points in geometric space. In trigonometry, we use these measurements to describe the size of angles and their rotations. Breaking down a circle into these equal parts makes it easier to understand and calculate relationships between angles in various mathematical problems.

Trigonometry Basics

Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles. A fundamental skill in trigonometry involves working with angles, particularly in problems related to rotations and circles.

• Angles can be expressed in degrees or radians, though degrees are more common in basic problems.
• Understanding sine, cosine, and tangent functions helps describe the relationships between the angles and lengths in right triangles.

In the context of circular movements, trigonometry also explores how these angles sit along the circle's path. The concept of trigonometry extends our ability to describe scenarios ranging from simple geometric shapes to the orbits of planets. With its roots in geometry, trigonometry empowers students to explore mathematical concepts involving waves, cycles, and rotational motion.

Coterminality

Coterminal angles are those that share the same terminal side, essentially ending at the same spot on the circle, but they might involve different amounts of rotation. For example, if you start at the positive x-axis and rotate all the way around by 360 degrees, you end up back at your starting point, where a 0-degree angle would be. To find a coterminal angle between 0° and 360° from any given angle, you either add or subtract 360° until the angle falls within the desired range.

• Given angle: 1270°
• Subtract 360° repeatedly: 1270° - 3 x 360° = 190°
• New angle: 190° is within 0° to 360°

This means that a 190-degree rotation lands you in the same position as a 1270-degree rotation, proving their coterminality. This technique is crucial in trigonometry because it helps simplify problems by reducing large angles to more easily manageable representations within the standard range of a circle.