Question

1. If cscθ=5/3 then secθ=

a. 25/16
b. 5/4
c. 16/25
d. 4/5

2. Simplify the expression
(cos x) (sec x) - (sin2 x)
a. cos x
b. cos2 x
c. sec2 x

3. If θ is in Quadrant IV and sinθ=-5/4 then secθ=
a. -13/12
b. -12/3
c. 12/13
d. 13/12

Answer 1

(1)

b.
`sec(\theta)=(5)/(4)`

(2)

b.
`cos^2(x)`

(3)

`sec(\theta)=(13)/(12)`

Explanation:

(1)

we are given

`csc(\theta)=(5)/(3)`

we can use triangle method

we know that

csc=hyp/ opposite

so, hyp=5

opposite =3

now, we can find adjacent

we can use Pythagoras theorem

`hyp^2=opp^2+adj^2`

`5^2=3^2+adj^2`

`adj=4`

now, we can find sec

`sec(\theta)=(5)/(4)`

(2)

we are given

`(cosx)(secx)-sin^2(x)`

we can simplify it

`(cosx)* ((1)/(cosx))-sin^2(x)`

`1-sin^2(x)`

now, we can replace 1 as sin^2x +cos^2x

we get

`sin^2(x)+cos^2(x)-sin^2(x)`

`=cos^2(x)`

(3)

we are given

θ is in Quadrant IV

`sin(\theta)=(-5)/(13)`

we know that

sin =opp/hyp

so, opp=5

hyp=13

now, we can use Pythagoras theorem

`hyp^2=opp^2+adj^2`

now, we can plug values

`13^2=5^2+adj^2`

`adj=12`

sec=hyp/adj

so, we get

`sec(\theta)=(13)/(12)`