Question

Evaluate tan 75° and tan 165° without using a calculator or tables.

The answer is 2 +
`√(3)`, -2 +
`√(3)`

Answer 1

`\tan 75^(\circ)=2+√(3)`

`\tan 165^(\circ)=-2+√(3)`

Explanation:

Tangent addition formula

`\boxed{\tan (A + B)=(\tan A + \tan B)/(1 - \tan A \tan B)}`

To evaluate tan 75°, we can use the tangent addition formula.

If A + B = 75° and 30° + 45° = 75°, then:

• A = 30°
• B = 45°

Substitute these values into the formula:

`\implies\tan (30^(\circ) +45^(\circ))=(\tan 30^(\circ) + \tan 45^(\circ))/(1 - \tan 30^(\circ) \tan 45^(\circ))`

`\textsf{As\;\;$\tan 30^(\circ) = (√(3))/(3)$\;\;and\;\;$\tan 45^(\circ)= 1$,\;substitute\;these\;values\;into\;the\;formula:}`

`\implies\tan (30^(\circ) +45^(\circ))=((√(3))/(3) + 1)/(1 - (√(3))/(3) \cdot 1)`

Simplify:

`\implies\tan (30^(\circ) +45^(\circ))=((√(3))/(3) + 1)/(1 - (√(3))/(3))`

`\implies\tan (30^(\circ) +45^(\circ))=((√(3))/(3) + (3)/(3))/((3)/(3) - (√(3))/(3))`

`\implies\tan (30^(\circ) +45^(\circ))=((3+√(3))/(3))/((3-√(3))/(3))`

`\implies\tan (30^(\circ) +45^(\circ))=(3+√(3))/(3-√(3))`

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:

`\implies\tan (30^(\circ) +45^(\circ))=((3+√(3))(3+√(3)))/((3-√(3))(3+√(3)))`

Simplify:

`\implies\tan (30^(\circ) +45^(\circ))=(9+6√(3)+3)/(9-3)`

`\implies\tan (30^(\circ) +45^(\circ))=(12+6√(3))/(6)`

`\implies\tan (30^(\circ) +45^(\circ))=(12)/(6)+(6√(3))/(6)`

`\implies\tan (30^(\circ) +45^(\circ))=2+√(3)`

`\implies\tan 75^(\circ)=2+√(3)`

`\hrulefill`

To evaluate tan 165°, we can use the tangent addition formula.

If A + B = 165° and 120° + 45° = 165°, then:

• A = 120°
• B = 45°

Substitute these values into the formula:

`\implies\tan (120^(\circ) +45^(\circ))=(\tan 120^(\circ) + \tan 45^(\circ))/(1 - \tan 120^(\circ) \tan 45^(\circ))`

As tan 120° = -√3 and tan 45° = 1, substitute these values into the formula:

`\implies\tan (120^(\circ) +45^(\circ))=(-√(3)+ 1)/(1 - (-√(3)) \cdot 1)`

Simplify:

`\implies\tan (120^(\circ) +45^(\circ))=(1-√(3))/(1 +√(3))`

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:

`\implies\tan (120^(\circ) +45^(\circ))=((1-√(3))(1-√(3)))/((1 +√(3))(1-√(3)))`

Simplify:

`\implies\tan (120^(\circ) +45^(\circ))=(1-2√(3)+3)/(1-3)`

`\implies\tan (120^(\circ) +45^(\circ))=(4-2√(3))/(-2)`

`\implies\tan (120^(\circ) +45^(\circ))=(4)/(-2)-(2√(3))/(-2)`

`\implies\tan (120^(\circ) +45^(\circ))=-2+√(3)`

`\implies\tan 165^(\circ)=-2+√(3)`