Question

It is given that α is anacute angle and that tan α = `sqrt(5)/"2"` . Answer the follwoing question!?

It is given that α is an acute angle and that tan α = `sqrt(5)/"2"`

(a) Draw a right-angled triangle showing this information.

(b) Use Pythagoras’ theorem to find the length of the unknown side.

(c) Hence:
(i) write down the exact values of sin α and cos α,

(ii) show that `sin^2α + cos^2α = 1`

Answer 1

Please see below.

Explanation:

.

`color(red)((a))`

`tanalpha=("Opposite")/("Adjacent")=sqrt5/2`

`color(red)((b))`

`(BC)^2=(AB)^2+(AC)^2`

`(BC)^2=2^2+(sqrt5)^2=4+5=9`

`BC=sqrt9=3`

`color(red)((c))`

`color(red)((i))`

`sinalpha=("Opposite")/("Hypotenuse")=sqrt5/3`

`cosalpha=("Adjacent")/("Hypotenuse")=2/3`

`color(red)((ii))`

`sin^2alpha=(sqrt5/3)^2=5/9`

`cos^2alpha=(2/3)^2=4/9`

`sin^2alpha+cos^2alpha=5/9+4/9=9/9=1`