How do you find the exact values of tan 67.5° using the half angle formula?

To find the exact value of tan 67.5°, we can use the half-angle formula for tangent. The half-angle formula states:

Half-Angle Formula

The formula is given by:

tan(θ/2) = √((1 - cos(θ)) / (1 + cos(θ)))

Choosing the Angle

In this case, we can express 67.5° as half of 135°:

θ = 135°

Finding cos(135°)

Next, we need to calculate cos(135°). Since 135° is in the second quadrant:

• cos(135°) = -cos(45°)
• cos(45°) = √2/2

Thus, cos(135°) = -√2/2.

Applying the Half-Angle Formula

Now, substitute cos(135°) into the half-angle formula:

tan(67.5°) = √((1 - (-√2/2)) / (1 + (-√2/2)))

Simplifying the Expression

Let's simplify the expression step by step:

• 1 - (-√2/2) = 1 + √2/2
• 1 + (-√2/2) = 1 - √2/2

So, we have:

tan(67.5°) = √((1 + √2/2) / (1 - √2/2))

Final Calculation

To simplify further, multiply the numerator and denominator by 2:

tan(67.5°) = √((2 + √2) / (2 - √2))

This gives you the exact value of tan 67.5°. You can evaluate this expression for a numerical approximation if needed, but this is the exact form.