Solve the equation for t on the interval left bracket 0 comma 2 pi right parenthesis.
4 sine t minus 4 cosine tequals4
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
tequalsnothing
  
​(Simplify your answer. Type your answer in radians. Type an exact answer in terms of pi . Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)
B.
There is no solution.

Let's solve the equation:
\[
4 \sin(t) - 4 \cos(t) = 4.
\]

### Step 1: Simplify the equation
Divide the entire equation by 4:
\[
\sin(t) - \cos(t) = 1.
\]

### Step 2: Use a trigonometric identity
Rewrite \(\sin(t) - \cos(t)\) using a single sine function. We can express this in the form \( R \sin(t + \theta) \), where \( R = \sqrt{1^2 + (-1)^2} = \sqrt{2} \), and \( \theta = \tan^{-1}\left(-\frac{1}{1}\right) = -\frac{\pi}{4} \).

Thus, we can rewrite the equation as:
\[
\sqrt{2} \sin\left(t - \frac{\pi}{4}\right) = 1.
\]

### Step 3: Solve for \( t \)
Divide by \(\sqrt{2}\):
\[
\sin\left(t - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}.
\]

The solutions for \(\sin\left(t - \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\) within one period \(0 \leq t < 2\pi\) are:
\[
t - \frac{\pi}{4} = \frac{\pi}{4} \quad \text{and} \quad t - \frac{\pi}{4} = \pi - \frac{\pi}{4}.
\]

### Step 4: Solve for \( t \)
1. \( t = \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2} \).
2. \( t = \frac{\pi}{4} + \frac{3\pi}{4} = \pi \).

Thus, the solutions are:
\[
t = \frac{\pi}{2}, \pi.
\]

The correct answer is:
**A. \( t = \frac{\pi}{2}, \pi \).**

Would you like further details on the steps or have any questions?

### Related Questions:
1. How do you solve trigonometric equations involving both sine and cosine?
2. What are the steps for converting a linear combination of sine and cosine into a single trigonometric function?
3. How do you find solutions to trigonometric equations over a specific interval?
4. What is the general solution for \(\sin(x) = a\) over any interval?
5. How does the sine function behave over the interval \(0\) to \(2\pi\)?

### Tip:
When solving trigonometric equations, rewriting the equation in terms of a single trigonometric function often simplifies finding the solutions.

Solve the equation for t on the interval [0, 2π): 4 sin(t) - 4 cos(t) = 4. Select the correct choice and, if necessary, fill in the answer box to complete your choice.

Learn to solve the trigonometric equation 4 sin(t) - 4 cos(t) = 4 on the interval [0, 2π). This problem covers key concepts in trigonometry, including using single sine form, solving trigonometric equations, and understanding interval solutions. Step-by-step guidance is provided, ideal for high school students in Grades 10-12.

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