Question

How do i get the exact value of sec 225°?

Answer 1

`sec225°` = `-sqrt2`

Explanation:

Apply the identity `sectheta=1/(costheta)`

`sec225° = 1/(cos225°)`

`1/(cos225°)` = `1/(cos(180°+45°)`

Use the angle sum identity `cos(a+b)=(cosa ⋅ cosb)-sina ⋅ sinb)`
where `a=180°` and `b=45°`

`1/(cos(180°+45°)`=`1/((cos180° ⋅ cos45°) - (sin180° ⋅ sin45°))`

`cos180°=-1`

`cos45°=(√2) / 2`

`sin180°=0`

`sin45°=(√2) / 2`

`1/((cos180° ⋅ cos45°) - (sin180° ⋅ sin45°))`=`1/((-(√2) / 2)-0`

`1/((-(√2) / 2)`=`-2/sqrt2`

Rearrange the equation so there is no square root in the denominator

`-2/sqrt2` ⋅ `sqrt2/sqrt2` = `-(2sqrt2)/2` = `-sqrt2`

Answer 2

`- sqrt2`

Explanation:

Use trig identity:
cos (a + b) = cos a.cos b - sin a. sin b
In this case -->
cos 225 = cos (180 + 45) = cos 180.cos 45 - sin 180.sin 45.
Since: sin 180 = 0; cos 180 = -1, and `cos 45 = sqrt2/2`, therefor:
`cos 225 = (-1)sqrt2/2 = - sqrt2/2`
`sec 225 = 1/(cos 225) = - 2/(sqrt2) = - sqrt2`
Check by calculator.
cos 225 = - 0.707 --> sec 225 = 1/0.707 = 1.414
`sec 225 = -sqrt2 = - 1.414`. Proved.