Use pythagorean identity to find sin (0) if 0 is in quadrant iii

Given that:

TanФ = 5√11 ÷ 11

Using SOH CAH TOA

Sin Ф = opp/hyp

CosФ = adj/hyp

TanФ = opp/adj

Therefore,

From the given identity we can say that:

Opposite = 5√11

Adjacent = 11

Using Pythagoras formula:

a² + b² = c²

Where c is the hypotenuse, a and b are the opposite and adjacent.

We can now find the hypotenuse.

(5√11)² + 11² = c²

(25 × 11) + 121 = c²

275 + 121 = c²

396 = c²

Taking square root on both sides,

we get:

c = √396

c = 6√11

Now, we have all the three sides and again using SOH CAH TOA, we can figure out the exact value of Sin Ф.

Sin Ф = Opp/Hyp

Sin Ф = 5√11 ÷ 6√11

Canceling the like terms,

we get:

Sin Ф = 5/6

Now, we know that, in the third quadrant, only TanФ is positive and the rest of the trigonometric values are negative, we will have the negative value of the opposite side.

Hence,

 Sin Ф = - 5/6

we find the value of sin(θ), as θ is in third quadrant.

Given that

tan(θ)=

By Pythagorean identity,

1+tan²θ=sec²θ

1+(5√11/11)²=sec²θ

sec²θ=36/11

secθ=-6√11/11 (we omit the + value and chose the - value because in third quadrant secθ is negative).

Now,

cosθ= 1/secθ

=1/ -6√11/11

=-√11/6

use

sin²θ+cos²θ=1

sin²θ=1-cos²θ

sin²θ=1-(-√11/6)²

sin²θ=25/36

sinθ = -5/6 (we omit the + value and chose the - value because in third quadrant secθ is negative).

Option A is correct.