3π/4

Definition

3π/4 is an angle measurement in radians that corresponds to 135 degrees. This angle lies in the second quadrant of the unit circle, where the sine function is positive and the cosine function is negative. Understanding 3π/4 is crucial for interpreting trigonometric functions and their values at specific angles.

5 Must Know Facts For Your Next Test

1. 3π/4 radians is equivalent to 135 degrees, which can be calculated using the conversion formula: degrees = radians × (180/π).
2. In the unit circle, the coordinates for 3π/4 are (-√2/2, √2/2), indicating that sine is positive and cosine is negative at this angle.
3. The tangent of 3π/4 is -1, which can be derived from the sine and cosine values: tangent = sine/cosine.
4. This angle is significant in various applications, such as in solving equations involving trigonometric functions and analyzing periodic behavior.
5. When graphing trigonometric functions, 3π/4 plays an important role in determining the behavior and symmetry of sine and cosine waves.

Review Questions

• How does the position of 3π/4 on the unit circle affect the values of sine and cosine?
  • At 3π/4 on the unit circle, which is located in the second quadrant, the sine value is positive while the cosine value is negative. This positioning reflects how angles are measured and how they correspond to specific points on the unit circle. The coordinates for this angle are (-√2/2, √2/2), highlighting these characteristic values for sine and cosine.
• Calculate and explain the tangent value for 3π/4, and why it holds that value based on sine and cosine.
  • The tangent of 3π/4 can be calculated by dividing its sine value by its cosine value. Since sine(3π/4) = √2/2 and cosine(3π/4) = -√2/2, we find that tangent(3π/4) = (√2/2) / (-√2/2) = -1. This negative value indicates that in this quadrant, where sine is positive and cosine is negative, tangent will always yield a negative result.
• Discuss how understanding 3π/4 enhances your ability to solve trigonometric equations involving angles in various quadrants.
  • Grasping the significance of 3π/4 helps in solving trigonometric equations as it provides insights into how angles behave differently across quadrants. Since 3π/4 is located in the second quadrant where sine remains positive while cosine turns negative, recognizing this helps predict function values when solving equations. This knowledge enables more accurate interpretations of solutions across all quadrants, thereby enhancing problem-solving strategies with trigonometric identities and functions.

Related terms

Radians: A unit of angular measure where one radian is the angle formed when the arc length equals the radius of the circle.

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

Trigonometric Functions:

Functions that relate the angles of a triangle to the lengths of its sides, including sine, cosine, and tangent.