Math Problem Statement

Sur le cercle trigonométrique suivant, donner le réel dont le point A est l'image.

Solution

The question asks to determine the real value associated with the point AA on the given unit circle.

Solution:

From the diagram:

  • The point AA is located in the first quadrant.
  • The coordinates of AA are (32,12)\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right).

In the unit circle, the x-coordinate represents cos(θ)\cos(\theta), and the y-coordinate represents sin(θ)\sin(\theta). Therefore, we need to determine the angle θ\theta such that: cos(θ)=32,sin(θ)=12.\cos(\theta) = \frac{\sqrt{3}}{2}, \quad \sin(\theta) = \frac{1}{2}.

This corresponds to the angle θ=π6\theta = \frac{\pi}{6} (or 30 degrees), since: cos(π6)=32,sin(π6)=12.\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}.

Final Answer:

The real number associated with the point AA is: θ=π6.\theta = \frac{\pi}{6}.

Let me know if you would like more detailed steps or additional help.


Related Questions:

  1. What is the process to calculate the angle for a different point on the unit circle?
  2. How are the sine and cosine functions defined on the unit circle?
  3. How does the unit circle relate to trigonometric identities?
  4. What are the coordinates of AA if rotated by π/2\pi/2 radians?
  5. Can you determine the angle for (32,12)\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)?

Tip:

Always ensure that you analyze whether the point is in the correct quadrant to verify the sign of sine and cosine.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Unit Circle
Angles and Radians

Formulas

cos(θ) = x-coordinate
sin(θ) = y-coordinate

Theorems

Definition of the Unit Circle

Suitable Grade Level

Grades 10-12