l1 * cos(q1) + l2 * cos(q1 + q2) = a
l1 * sin(q1) + l2 * sin(q1 + q2) = b

The given system of equations:

\[
l_1 \cos(q_1) + l_2 \cos(q_1 + q_2) = a
\]
\[
l_1 \sin(q_1) + l_2 \sin(q_1 + q_2) = b
\]

appears to represent the forward kinematics of a 2-link robotic arm, where:

- \( l_1 \) and \( l_2 \) are the lengths of the two links.
- \( q_1 \) and \( q_2 \) are the joint angles.
- \( a \) and \( b \) are the coordinates of the end-effector (the position of the tip of the arm) in the 2D plane.

### Approach to solve for \( q_1 \) and \( q_2 \):
We can approach solving this system by recognizing that the two equations represent the Cartesian coordinates of the end-effector. Solving for the angles \( q_1 \) and \( q_2 \) involves using inverse trigonometric functions.

### Step-by-step plan:
1. **First equation for angle \( q_1 \):**
   Add the two equations into a form where the trigonometric identities can be used.

2. **Use geometric relationships:**
   Consider manipulating the equations by squaring them and adding them together to simplify the sine and cosine terms.

Would you like me to go ahead and solve the system step by step to find \( q_1 \) and \( q_2 \)?

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#### Expanding the understanding:
1. How do sine and cosine relate to vector representation?
2. What are inverse trigonometric functions?
3. What real-world applications rely on solving such trigonometric systems?
4. How are forward kinematics used in robotics?
5. What are other ways to model joint angles and link positions?

**Tip**: Always remember that trigonometric systems can often be simplified by using identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \).

Discover how to solve a system of trigonometric equations representing the forward kinematics of a 2-link robotic arm. This problem involves using inverse trigonometric functions and geometric relationships to determine the joint angles q1 and q2. Suitable for advanced high school or undergraduate students.

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