If sin(θ) = 4/5 and sin(φ) = 12/13, evaluate tan(θ - φ).

Steps to Solve

1. Identify the given values for sine and cosine

From the problem, we have:

• ( \sin \theta = \frac{4}{5} )
• ( \sin \phi = \frac{5}{13} )

2. Calculate corresponding cosine values using the Pythagorean identity

Using the identity ( \sin^2 + \cos^2 = 1 ):

For ( \theta ): $$ \cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5} $$

For ( \phi ): $$ \cos \phi = \sqrt{1 - \sin^2 \phi} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13} $$

3. Use the tangent difference formula

The tangent difference formula is given by: $$ \tan(\theta - \phi) = \frac{\tan \theta - \tan \phi}{1 + \tan \theta \tan \phi} $$

First, we must find ( \tan \theta ) and ( \tan \phi ):

• ( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} )
• ( \tan \phi = \frac{\sin \phi}{\cos \phi} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12} )

4. Substitute ( \tan \theta ) and ( \tan \phi ) into the formula

Using the values we calculated: $$ \tan(\theta - \phi) = \frac{\frac{4}{3} - \frac{5}{12}}{1 + \left(\frac{4}{3}\right) \left(\frac{5}{12}\right)} $$

5. Simplify the numerator and denominator

Calculating the numerator: $$ \frac{4}{3} - \frac{5}{12} = \frac{16}{12} - \frac{5}{12} = \frac{11}{12} $$

Calculating the denominator: $$ 1 + \left(\frac{4}{3} \cdot \frac{5}{12}\right) = 1 + \frac{20}{36} = 1 + \frac{5}{9} = \frac{9}{9} + \frac{5}{9} = \frac{14}{9} $$

6. Final calculation of ( \tan(\theta - \phi) )

Now substituting back: $$ \tan(\theta - \phi) = \frac{\frac{11}{12}}{\frac{14}{9}} = \frac{11}{12} \cdot \frac{9}{14} = \frac{99}{168} = \frac{33}{56} $$