Find all solutions t between 360 and 720, inclusive:

(a) cos t = sin t

To solve the equation \( \cos t = \sin t \) for \( t \) in the interval \([360^\circ, 720^\circ]\), we proceed step by step.

### Step 1: Use a trigonometric identity

We can divide both sides of the equation by \( \cos t \) (as long as \( \cos t \neq 0 \)):

\[
\frac{\cos t}{\cos t} = \frac{\sin t}{\cos t}
\]

This simplifies to:

\[
1 = \tan t
\]

### Step 2: Solve for \( t \)

Now, we need to find the general solution to \( \tan t = 1 \). The tangent function equals 1 at:

\[
t = 45^\circ + 180^\circ n
\]

where \( n \) is an integer. This represents the general solution for all values of \( t \).

### Step 3: Find solutions in the given interval

Now we restrict \( t \) to the interval \([360^\circ, 720^\circ]\). We plug values of \( n \) to find the specific solutions.

1. For \( n = 2 \):
   \[
   t = 45^\circ + 180^\circ(2) = 45^\circ + 360^\circ = 405^\circ
   \]

2. For \( n = 3 \):
   \[
   t = 45^\circ + 180^\circ(3) = 45^\circ + 540^\circ = 585^\circ
   \]

3. For \( n = 4 \):
   \[
   t = 45^\circ + 180^\circ(4) = 45^\circ + 720^\circ = 765^\circ
   \]
   This value is outside the interval \([360^\circ, 720^\circ]\), so it is not included.

### Final Answer:

The solutions for \( t \) in the interval \([360^\circ, 720^\circ]\) are:
\[
t = 405^\circ, 585^\circ
\]

Would you like more details on any step? Here are 5 related questions to consider:

1. How does the tangent function behave across intervals of 180 degrees?
2. Why did we divide by \( \cos t \) and not \( \sin t \)?
3. What happens when both \( \cos t \) and \( \sin t \) are zero?
4. How can we express solutions to other trigonometric equations using general formulas?
5. What are the key identities involving \( \tan t \)?

**Tip:** When solving trigonometric equations, always check for restrictions such as dividing by zero or undefined values.

Find all solutions t between 360 and 720, inclusive: (a) cos t = sin t

Learn how to solve the trigonometric equation cos t = sin t and find the solutions for t between 360° and 720°. This detailed guide covers key trigonometric concepts, including using the identity tan t = 1 and periodicity of the tangent function. Suitable for students in Grades 10-12.

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