To solve this problem, we need to express $7 + 7i$ in trigonometric (polar) form, which is given as:

$r(\cos\theta + i\sin\theta)$

Step-by-step solution:

1. Find the modulus $r$: The modulus $r$ of the complex number is: $r = \sqrt{a^2 + b^2}$ Here, $a = 7$ and $b = 7$. Substituting: $r = \sqrt{7^2 + 7^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}.$

2. Find the argument $\theta$: The argument $\theta$ is the angle made with the positive real axis, calculated using: $\tan\theta = \frac{b}{a}.$ Substituting $a = 7$ and $b = 7$: $\tan\theta = \frac{7}{7} = 1.$ For $\tan\theta = 1$, $\theta$ is $45^\circ$ (in the first quadrant).

3. Write the trigonometric form: Using $r = 7\sqrt{2}$ and $\theta = 45^\circ$, the trigonometric form is: $7 + 7i = 7\sqrt{2} (\cos 45^\circ + i\sin 45^\circ).$

Final Answer:

$7 + 7i = 7\sqrt{2} (\cos 45^\circ + i\sin 45^\circ).$

Would you like further clarification or assistance with this topic?

Related Questions:

1. How do you convert other complex numbers into polar form?
2. What is the importance of the modulus in polar representation?
3. How can we convert polar form back to rectangular form?
4. What are the applications of trigonometric form in mathematics?
5. How does the angle $\theta$ relate to the quadrant of the complex number?

Tip:

Always check the quadrant of your complex number before determining the angle $\theta$ to avoid incorrect signs.