45°

Definition

45° is an angle that represents one-eighth of a full circle, or 1/8 of 360°. It is a fundamental angle in trigonometry and is often used as a reference point in the unit circle.

5 Must Know Facts For Your Next Test

1. In the unit circle, the coordinates for the point at 45° are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.
2. The angle of 45° is equal to $\frac{\pi}{4}$ radians, as there are $2\pi$ radians in a full circle.
3. The 45° angle is located in the first quadrant of the unit circle, where both the x- and y-coordinates are positive.
4. The trigonometric functions of 45° are: $\sin(45°) = \frac{\sqrt{2}}{2}$, $\cos(45°) = \frac{\sqrt{2}}{2}$, and $\tan(45°) = 1$.
5. The 45° angle is often used as a reference point in various mathematical and scientific applications, such as in the design of electrical circuits and the analysis of mechanical systems.

Review Questions

• Explain the significance of the 45° angle in the context of the unit circle.
  • The 45° angle is a fundamental angle in the unit circle, as it represents one-eighth of a full circle. This angle is significant because it serves as a reference point for defining the trigonometric functions, such as sine, cosine, and tangent. The coordinates of the point on the unit circle at 45° are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, which are important values used in various mathematical and scientific applications. Additionally, the 45° angle is located in the first quadrant of the unit circle, where both the x- and y-coordinates are positive, making it a useful reference point for understanding the behavior of trigonometric functions.
• Describe how the 45° angle is related to the concept of radians.
  • The 45° angle is closely related to the concept of radians, which is another unit of angle measurement. Specifically, the 45° angle is equal to $\frac{\pi}{4}$ radians, as there are $2\pi$ radians in a full circle. This relationship between degrees and radians is important for understanding the unit circle and the behavior of trigonometric functions. Knowing the radian measure of the 45° angle allows for easier conversion between the two units of angle measurement, which is essential in many mathematical and scientific applications.
• Analyze the trigonometric function values for the 45° angle and explain their significance.
  • The trigonometric function values for the 45° angle are: $\sin(45°) = \frac{\sqrt{2}}{2}$, $\cos(45°) = \frac{\sqrt{2}}{2}$, and $\tan(45°) = 1$. These values are significant because they represent the coordinates of the point on the unit circle at 45°, which is located in the first quadrant. The fact that the sine and cosine values are both equal to $\frac{\sqrt{2}}{2}$ indicates that the 45° angle is a special angle where the sine and cosine values are equal. Additionally, the tangent value of 1 for the 45° angle is also a notable characteristic, as it represents the slope of the line tangent to the unit circle at that point.

Related terms

Unit Circle:

A circle with a radius of 1 unit, used to define trigonometric functions and relate angles to their corresponding coordinates on the circle.

Quadrant:

One of the four equal sections of a circle or coordinate plane, divided by the x- and y-axes.

Radian:

A unit of angle measurement that represents the ratio of an arc length to the radius of the circle, where 1 radian is equal to approximately 57.3 degrees.