Question

If tan68° = 1/m, express the following in terms of m:

1. tan22°

Answer 1

We can use the tangent addition formula to find an expression for tan 22° in terms of tan 68°:

tan(68° - 46°) = (tan 68° - tan 46°)/(1 + tan 68° * tan 46°)

tan 22° = (tan 68° - tan 46°)/(1 + tan 68° * tan 46°)

We know that tan 68° = 1/m, so we can substitute:

tan 22° = (1/m - tan 46°)/(1 + (1/m) * tan 46°)

Therefore, tan 22° can be expressed in terms of m as:

tan 22° = (1/m - tan 46°)/(1 + tan 46°/m)

Answer 2

Using the identity tan(3θ) = (3tanθ - tan^3θ) / (1 - 3tan^2θ), we can say that:

tan(3 * 22°) = (3tan22° - tan^3(22°)) / (1 - 3tan^2(22°))

Solving for tan(22°), we get:

tan(22°) = (3tan(3 * 22°) - tan^3(3 * 22°)) / (3tan^2(3 * 22°) - 1)

Now, using the identity tan(3θ) = (3tanθ - tan^3θ) / (1 - 3tan^2θ), we can also say that:

tan(3 * 68°) = (3tan68° - tan^3(68°)) / (1 - 3tan^2(68°))

Given that tan68° = 1/m, we can substitute this value into the equation above to get:

tan(3 * 68°) = (3/m - 1/m^3) / (1 - 3/m^2)

Simplifying this expression, we get:

tan(3 * 68°) = (3 - m^2) / (m^3 - 3m)

Now, substituting the value of tan(3 * 22°) = (3 - √3) / (3√3 - 1) into the equation for tan(22°), we get:

tan(22°) = (3(3 - √3)/(3√3 - 1) - (3 - √3)^3/(3√3 - 1)^3) / (3(3 - √3)/(3√3 - 1)^2 - 1)

Simplifying this expression, we get:

tan(22°) = (27 - 9√3 - 27√3 + 9√3) / (27√3 - 9 - 3√3 + 1)

tan(22°) = (18 - 18√3) / (26√3 - 8)

Therefore, we can express tan(22°) in terms of m as:

tan(22°) = (18 - 18√3) / (26√3