Give one positive and one negative coterminal angle for 135°
The Correct answer and Explanation is:
To find coterminal angles for a given angle, we can add or subtract multiples of 360°. Coterminal angles are angles that share the same terminal side when drawn in standard position, meaning they look the same on a coordinate plane even though they may have different measures.
Finding Coterminal Angles for 135°
- Positive Coterminal Angle: To find a positive coterminal angle, we add 360° to 135°:135°+360°=495°135° + 360° = 495°135°+360°=495°Thus, 495° is a positive coterminal angle to 135°.
- Negative Coterminal Angle: To find a negative coterminal angle, we subtract 360° from 135°:135°−360°=−225°135° – 360° = -225°135°−360°=−225°Hence, -225° is a negative coterminal angle to 135°.
Explanation
The concept of coterminal angles is fundamental in trigonometry and helps in understanding periodic functions. Angles are measured in degrees or radians, and when we talk about coterminal angles, we’re essentially looking at different ways to represent the same direction or rotation in a circle.
In our example, starting from the positive angle of 135°, which lies in the second quadrant (where sine is positive and cosine is negative), adding 360° simply completes one full rotation and lands us back at the same angle, thus resulting in 495°. This shows that angles can repeat every full rotation, hence the infinite possibilities of coterminal angles.
On the other hand, subtracting 360° gives us -225°, which indicates a clockwise rotation from the initial position. The negative angle also terminates at the same position as 135°, demonstrating the cyclic nature of angular measurements.
Understanding coterminal angles is essential for solving various problems in trigonometry, such as simplifying trigonometric functions or solving equations that involve angle measures. By working with coterminal angles, one can often make calculations easier by reducing angles to their simplest forms within the standard range (0° to 360°).