Understanding Coterminal Angles

Coterminal angles are angles in standard position (angles with the initial side on the positive x-axis) that share the same terminal side. This means they represent the same position on the unit circle but might involve different amounts of rotation.

You can find coterminal angles by adding or subtracting multiples of 360 degrees (a full circle) to the original angle. The general formula for coterminal angles is:

$$ \theta_{coterminal} = \theta_{original} + n \cdot 360^\circ $$

where n is any integer (..., -2, -1, 0, 1, 2, ...).

Finding the Equivalent Angle to 135°

We are given the angle $135^\circ$ and need to find which of the options is coterminal with it. We can test each option using the formula:

Option 1: -225°

Let's see if we can get -225° by adding or subtracting 360° from 135°:

Is $-225^\circ = 135^\circ + n \cdot 360^\circ$ for some integer n?

Subtracting $135^\circ$ from both sides:

$$ -225^\circ - 135^\circ = n \cdot 360^\circ $$

$$ -360^\circ = n \cdot 360^\circ $$

Dividing by $360^\circ$:

$$ n = -1 $$

Since n = -1 is an integer, -225° is coterminal with 135°.

Option 2: -45°

Is $-45^\circ = 135^\circ + n \cdot 360^\circ$?

$$ -45^\circ - 135^\circ = n \cdot 360^\circ $$

$$ -180^\circ = n \cdot 360^\circ $$

$$ n = -180^\circ / 360^\circ = -1/2 $$

Since n is not an integer, -45° is not coterminal with 135°.

Option 3: -135°

Is $-135^\circ = 135^\circ + n \cdot 360^\circ$?

$$ -135^\circ - 135^\circ = n \cdot 360^\circ $$

$$ -270^\circ = n \cdot 360^\circ $$

$$ n = -270^\circ / 360^\circ = -3/4 $$

Since n is not an integer, -135° is not coterminal with 135°.

Option 4: -315°

Is $-315^\circ = 135^\circ + n \cdot 360^\circ$?

$$ -315^\circ - 135^\circ = n \cdot 360^\circ $$

$$ -450^\circ = n \cdot 360^\circ $$

$$ n = -450^\circ / 360^\circ = -5/4 $$

Since n is not an integer, -315° is not coterminal with 135°.

Conclusion

By applying the definition of coterminal angles, we found that only -225° satisfies the condition $135^\circ + n \cdot 360^\circ$ with an integer value for n (specifically, $n = -1$). Therefore, -225° is the angle that is the same as 135° in terms of its position on the unit circle.