## Solving Hyperbolic Trigonometric Equations (2)

An equation of the form(1) may have none, one or two solutions, unlike for the equivalent ordinary trigonometric equation which may have many solutions.

We can solve equations of form (1) by substituting

and

Then multiply though byand we have a quadratic equation inwhich we solve by the normal method of substitution and factorisation or use of the quadratic formula.

Example: Solve the equation

After simplification and collection of like terms this becomes

Multiply byto obtain

Substituteto obtain

This expression factorises to give

We set each factor equal to 0 and solve for y, then use the original substitution to solve for

Example: Solve the equation

After simplification and collection of like terms this becomes

Multiply byto obtain

Substituteto obtain

This expression does not factorise and we must use the quadratic formula,

to 3 sf.

which has no value so the only solution is