Question

How to solve cos (6 theta) + i sin (6 theta)?

Option 1: Use the Pythagorean trigonometric identity.
Option 2: Convert it to exponential form.
Option 3: Apply the double angle formula for sine.
Option 4: Simplify using the half-angle formula for cosine.

Answer 1

To solve cos (6 theta) + i sin (6 theta), you can convert it to exponential form using Euler's formula. The solution is e^(i 6 theta).

Step-by-step explanation:

To solve cos (6 theta) + i sin (6 theta), we can use Option 2: Convert it to exponential form.

1. Start by converting the trigonometric functions to their exponential form using Euler's formula: cos (theta) = (e^(i theta) + e^(-i theta))/2 and sin (theta) = (e^(i theta) - e^(-i theta))/(2i).
2. Apply the exponential form to the given equation: cos (6 theta) + i sin (6 theta) = (e^(i 6 theta) + e^(-i 6 theta))/2 + i(e^(i 6 theta) - e^(-i 6 theta))/(2i).
3. Simplify the equation: (e^(i 6 theta) + e^(-i 6 theta))/2 + i(e^(i 6 theta) - e^(-i 6 theta))/(2i) = (e^(i 6 theta) + e^(-i 6 theta))/2 + (e^(i 6 theta) - e^(-i 6 theta))/2.
4. Combine like terms: (e^(i 6 theta) + e^(-i 6 theta) + e^(i 6 theta) - e^(-i 6 theta))/2 = 2(e^(i 6 theta))/2.
5. Simplify further: 2(e^(i 6 theta))/2 = e^(i 6 theta).

Therefore, the solution to cos (6 theta) + i sin (6 theta) is e^(i 6 theta).