45°

Definition

45° is an angle that is commonly found in both the unit circle and trigonometric functions, representing one of the key angles in trigonometry. This angle is significant because it has unique properties, especially in relation to the coordinates on the unit circle, where both the x and y values are equal. Understanding 45° helps to establish a foundation for working with special triangles, particularly the isosceles right triangle, and is critical for calculating sine, cosine, and tangent values.

5 Must Know Facts For Your Next Test

1. On the unit circle, the coordinates for the angle 45° are $$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$.
2. The sine and cosine values for 45° are equal, both being $$\frac{\sqrt{2}}{2}$$.
3. The tangent of 45° equals 1 since it is the ratio of sine to cosine.
4. The angle 45° plays a crucial role in constructing special triangles, where the sides opposite the 45° angles are equal.
5. In radians, 45° is represented as $$\frac{\pi}{4}$$.

Review Questions

• How does the angle of 45° relate to the coordinates on the unit circle?
  • The angle of 45° corresponds to specific coordinates on the unit circle, which are $$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$. This means that both the x and y coordinates are equal at this angle, reflecting its unique relationship in trigonometry. This symmetry simplifies calculations for sine and cosine, making them both equal to $$\frac{\sqrt{2}}{2}$$.
• Explain how the properties of an isosceles right triangle relate to the 45° angle.
  • An isosceles right triangle has two angles measuring 45° and one right angle of 90°. The lengths of the sides opposite these 45° angles are equal, which means they represent equal ratios when applying trigonometric functions. This relationship allows for easier calculations using trigonometric ratios, particularly when determining side lengths or angles based on known values.
• Evaluate how understanding the angle 45° can assist in solving more complex trigonometric problems.
  • Understanding the angle 45° provides a foundational knowledge that simplifies many complex trigonometric problems. Since it is linked to specific coordinate values on the unit circle and has straightforward sine, cosine, and tangent values (both $$\frac{\sqrt{2}}{2}$$ for sine and cosine and 1 for tangent), these can be applied to solve problems involving larger angles or multiple transformations. Mastering this angle also aids in recognizing patterns in other angles related to it, enhancing overall problem-solving skills in trigonometry.

Related terms

Unit Circle: A circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions.

Isosceles Right Triangle: A triangle with two equal sides and a 90° angle, where each of the other angles measures 45°.

Trigonometric Ratios: Ratios derived from the angles and sides of triangles, including sine, cosine, and tangent.