Answer: The value of tan π/8 is 2 − √2.
To find tan π/8, we can use the tangent half-angle identity, which states that
tan (θ/2) = 1-cos(θ)/ 1+cos(θ)
In this case, θ = π/4. First, calculate cos (π/4), which is √2/2.
Now, apply the half-angle identity:
tan (π/8) = 1-√2/2 / 1+√2/2
After simplification, the result is 2 − √2. This value represents the tangent of one-eighth of a full circle and is often used in trigonometric calculations and identities.
Conclusion:
The value of tan(π/8) can be found using the tangent half-angle identity, which states that tan(θ/2) = (1 – cos(θ))/(1 + cos(θ)). Applying this identity with θ = π/4, and simplifying, we find that tan(π/8) equals 2 – √2. This value represents the tangent of one-eighth of a full circle and is useful in trigonometric calculations and identities
Some Related Questions:
How is the tangent half-angle identity used to find the value of tan(π/8)?
The tangent half-angle identity allows us to express the tangent of an angle in terms of the cosine of half that angle. By applying this identity and simplifying, we can find the value of tan(π/8) as 2 – √2.
Can you explain the significance of the value 2 – √2 in trigonometry?
The value 2 – √2 represents the tangent of π/8, which is an important angle in trigonometry. This value appears in various trigonometric calculations and identities, particularly those involving angles related to one-eighth of a full circle.