Question

See screenshot below​

Answer 1

cos theta = -1/ 2

Explanation:

sin theta = -sqrt(3)/2

Drawing a triangle

We know sin theta = opp/hyp

We can determine the adj side using the Pythagorean theorem

a^2 + b^2 = c^2

adj^2 + (-sqrt(3))^2 = 2^2

adj^2 +3 = 4

adj^2 = 4-3

adj ^2 =1

Taking the square root of each side

adj = 1

We know that since it is in the third quadrant the adj side is negative

adj = -1

cos theta = adj / hyp

cos theta = -1/ 2

Answer 2

Solution given:

Sin
`\theta_(1)=(-√(3))/(2)`

`(opposite)/(hypotenuse)=(-√(3))/(2)`

equating corresponding value

opposite=-
`√(3)`

hypoyenuse=2

adjacent=x

By using Pythagoras law

hypotenuse²=opposite²+adjacent²

2²=-
`√(3²)`+x²

4=3+x²

x²=4-3

x=
`√(1)=1`

x=-1

In third quadrant

Cos angle is negative

Cos
`\theta_(1)=(-adjacent)/(hypotenuse)`

Cos
`\theta_(1)=(-1)/(2)`