π/2

Definition

π/2, or 90 degrees, is a crucial angle in trigonometry representing the point where the sine function reaches its maximum value of 1 and the cosine function equals 0. This angle is significant because it marks a transition in the unit circle and is associated with various properties and behaviors of trigonometric functions, particularly tangent, cotangent, secant, and cosecant. Understanding π/2 is essential for analyzing the periodic nature of these functions and their graphs.

5 Must Know Facts For Your Next Test

1. At π/2 radians, the coordinates on the unit circle are (0, 1), reflecting that sine is at its peak while cosine hits zero.
2. The tangent function is undefined at π/2 since it involves dividing by cosine, which equals zero at this angle.
3. In graphs of tangent and cotangent functions, π/2 marks a vertical asymptote for the tangent curve.
4. For secant and cosecant functions, π/2 corresponds to the maximum point for cosecant and an undefined point for secant.
5. When converting degrees to radians, π/2 is equivalent to 90 degrees, illustrating an important connection between the two systems of measuring angles.

Review Questions

• How does the value of π/2 influence the graphs of tangent and cotangent functions?
  • The value of π/2 is pivotal because it represents a vertical asymptote for the tangent function. At this angle, tangent approaches infinity since it involves division by cosine, which is zero. This leads to significant changes in the behavior of the graph where it jumps from positive to negative infinity around this angle. Cotangent behaves oppositely; it equals zero at this point since it's defined as cosine over sine, which creates distinct graphical features in its periodic nature.
• Discuss the role of π/2 in relation to sine and cosine functions on the unit circle.
  • On the unit circle, π/2 corresponds to the coordinates (0, 1), where sine reaches its maximum value of 1. This point indicates that when you move a quarter turn counterclockwise from the positive x-axis, sine captures the height of 1 while cosine drops to 0. This relationship highlights how these two fundamental functions interact at critical angles, influencing their periodic nature and symmetries in their graphs.
• Evaluate how understanding π/2 can enhance your ability to solve trigonometric equations involving secant and cosecant.
  • Understanding π/2 allows you to identify key properties when solving equations involving secant and cosecant. At this angle, cosecant achieves its maximum value while secant remains undefined due to cosine being zero. Recognizing these behaviors helps anticipate solution sets and informs you about potential restrictions on variable values in your equations. This knowledge not only streamlines problem-solving but also deepens comprehension of how angles impact function behavior across trigonometric identities.

Related terms

Unit Circle: A circle with a radius of one centered at the origin of a coordinate plane, which is used to define sine, cosine, and tangent for all angles.

Sine Function: A fundamental trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right triangle, equal to 1 at π/2.

Cosine Function: A fundamental trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right triangle, equal to 0 at π/2.