To find tan 225° using the half angle formula, we start by recognizing that 225° is equal to 180° + 45°. This means we can express 225° as 2 * 112.5°, but for our purpose, we can also work with the properties of angles in the unit circle.
The formula for tangent using the half-angle is given by:
tan(θ/2) = ±√((1 – cos θ) / (1 + cos θ))
For this calculation, we’ll break it down. The angle θ here is 450° (which is 225° multiplied by 2 because we want the half). However, let’s approach it simpler by knowing that:
- In the second quadrant (where 225° lies), the tangent function is negative.
- We can use the reference angle. The angle 225° has a reference angle of 45°.
Now, we know:
tan 45° = 1
And since tangent is negative in the second quadrant, we get:
tan 225° = -tan 45° = -1
So, using the half-angle formula, we can confirm that:
tan 225° = -1
This is how you can find tan 225° using the half-angle formula and angle properties.