Math Problem Statement

Use the unit circle to find all values of θ between 0 and 2π for which sin(θ) = 1/2.

Solution

The problem asks to find all values of θ\theta between 00 and 2π2\pi for which sin(θ)=12\sin(\theta) = \frac{1}{2}, using the unit circle.

Solution:

We know from the unit circle that sin(θ)=12\sin(\theta) = \frac{1}{2} occurs at two specific angles within one full rotation (from 00 to 2π2\pi):

  1. θ=π6\theta = \frac{\pi}{6} (30 degrees)
  2. θ=5π6\theta = \frac{5\pi}{6} (150 degrees)

These angles are in the first and second quadrants, where the sine function is positive.

Final Answer:

The two values of θ\theta in the range 0θ<2π0 \leq \theta < 2\pi for which sin(θ)=12\sin(\theta) = \frac{1}{2} are: θ=π6,5π6\theta = \frac{\pi}{6}, \frac{5\pi}{6}

Would you like more details, or do you have any questions about this?

Here are 5 related questions to explore further:

  1. What is the value of sin(θ)\sin(\theta) for θ=7π6\theta = \frac{7\pi}{6}?
  2. How would you solve cos(θ)=12\cos(\theta) = \frac{1}{2} in the same interval?
  3. What are the angles where sin(θ)=12\sin(\theta) = -\frac{1}{2}?
  4. How do you find the cosine of these angles using the unit circle?
  5. Can you generalize the solution for sin(θ)=k\sin(\theta) = k, where kk is any value?

Tip: Memorizing common sine and cosine values for angles like π6,π4,π3,\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, and their multiples will help solve many trigonometric problems quickly.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Unit Circle
Sine Function

Formulas

sin(θ) = opposite / hypotenuse
Values of sine function for standard angles

Theorems

Unit Circle Theorem
Sine values for angles in quadrants

Suitable Grade Level

Grades 10-12