Question

When
tan= a 12/5
than evaluate cos2a​

Answer 1

The value of Cos 2a is - (
`(119)/(169)` ) .

Explanation:

Given as

Tan a =
`(12)/(5)`

Now, ∵ Tan Ф =
`(\textrm Perpendicular)/(\textrm Base)`

So, Tan a =
`(\textrm Perpendicular)/(\textrm Base)`

Or,
`(\textrm Perpendicular)/(\textrm Base)` =
`(12)/(5)`

Or, From Pythagoras Theorem

Hypotenuse² = perpendicular² + Base²

Or, Hypotenuse² = 12² + 5²

Or, Hypotenuse² = 144 + 25

Or, Hypotenuse² = 169

∴ Hypotenuse =
`√(169)` =
`\pm`13

Take Hypotenuse = 13

Now,∵ Cos 2Ф = Cos²Ф - Sin²Ф

So, Cos 2a = Cos²a - Sin²a

or , Cos 2a = 1 - 2 Sin²a ∵ Cos²a + Sin²a = 1

Or, Cos 2a = 1 - 2 ×(
`(\textrm Perpendicular)/(\textrm Hpotenuse)`)²

Or , Cos 2a = 1 - 2 ×(
`(\textrm 12)/(\textrm 13)`)²

Or, Cos 2a = 1 - 2 × (
`(144)/(169)`)

Or, Cos 2a = (
`(169-288)/(169)`)

∴ Cos 2a = - (
`(119)/(169)`)

Hence the value of Cos 2a is - (
`(119)/(169)`) . answer