(√2/2, √2/2)

Definition

The coordinates (√2/2, √2/2) represent a specific point on the unit circle where the angle is 45 degrees or $$\frac{\pi}{4}$$ radians. At this angle, both the sine and cosine values are equal, which means that the x-coordinate and y-coordinate of this point are the same. This relationship illustrates how angles in the first quadrant have positive sine and cosine values and highlights the symmetry of the unit circle.

5 Must Know Facts For Your Next Test

1. (√2/2, √2/2) corresponds to an angle of 45 degrees or $$\frac{\pi}{4}$$ radians in standard position on the unit circle.
2. At this angle, both sine and cosine values equal $$\frac{\sqrt{2}}{2}$$, which means that for any point at 45 degrees, both coordinates are equal.
3. The coordinates (√2/2, √2/2) are useful in determining values for trigonometric functions such as sine and cosine at key angles.
4. In terms of symmetry, (√2/2, √2/2) shows that angles in the first quadrant are positive for both sine and cosine, reflecting their values.
5. These coordinates help illustrate the concept of reference angles and can be used to find sine and cosine values for angles in other quadrants.

Review Questions

• How does the point (√2/2, √2/2) relate to the concept of symmetry in the unit circle?
  • The point (√2/2, √2/2) exemplifies symmetry in the unit circle because it shows that for angles of 45 degrees or $$\frac{\pi}{4}$$ radians in the first quadrant, both sine and cosine values are equal. This equality indicates that as you reflect over the line y = x, the coordinates remain consistent. This property highlights how trigonometric functions behave symmetrically within their respective quadrants.
• Explain how (√2/2, √2/2) can be used to find sine and cosine values for angles greater than 45 degrees.
  • By using (√2/2, √2/2) as a reference point on the unit circle at 45 degrees, we can derive sine and cosine values for larger angles by applying periodic properties. For example, angles such as 135 degrees or 225 degrees will have sine and cosine values derived from this point by considering their respective quadrants. The reference angle remains 45 degrees, but depending on its quadrant, sine may be positive or negative while cosine behaves accordingly.
• Evaluate how understanding (√2/2, √2/2) aids in grasping broader trigonometric concepts like identities or equations.
  • Understanding (√2/2, √2/2) provides a foundational knowledge of how sine and cosine interact within trigonometric identities. For instance, knowing that sin(45°) = cos(45°) = $$\frac{\sqrt{2}}{2}$$ helps reinforce identities like sin²(θ) + cos²(θ) = 1. This understanding allows students to manipulate and solve more complex trigonometric equations effectively by recognizing patterns based on these fundamental coordinates.

Related terms

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

Sine and Cosine: Sine is the ratio of the length of the opposite side to the hypotenuse in a right triangle, while cosine is the ratio of the adjacent side to the hypotenuse.

Radians: A unit of angular measure based on the radius of a circle, where one complete revolution is $$2\pi$$ radians.