Question

Given cos 0 = √3 /3 and sin 0<0 .
What is the value of sin 0 ?

Answer 1

The value of sin 0, given that cos 0 = √3 / 3 and sin 0 < 0, is calculated using the Pythagorean identity. After evaluating the equation sin² 0 = 2/3, we take the negative square root because sin 0 is less than 0, yielding sin 0 = -√(2/3).

Step-by-step explanation:

We are given that cos 0 = √3 / 3 and that sin 0 < 0. To find the value of sin 0, we should look at the Pythagorean identity, which tells us that sin² 0 + cos² 0 = 1. Replacing cos 0 with the given value, we then have sin² 0 + (√3 / 3)² = 1.

Calculating the square of √3 / 3, we get 3 / 9, which simplifies to 1/3. Substituting this into our Pythagorean identity gives us sin² 0 + 1/3 = 1. To find sin² 0, we subtract 1/3 from both sides of the equation to get sin² 0 = 1 - 1/3 = 2/3. Taking the square root of both sides, we remember that sin 0 is negative, so we choose the negative square root, and we find that sin 0 = -√(2/3).

Answer 2

Choice 4 down

Step-by-step explanation:

The cosine is positive and we are told that the sin is negative. That means that we are in QIV, where x is positive (cos) and y is negative (sin). If we draw a right triangle in this quadrant, we will label the x axis with a 1 (this is the side adjacent to the reference angle for cos), and the hypotenuse as square root of 3. The sin ratio is the side opposite the reference angle over the hypotenuse. That means we need to solve for the side opposite using Pythagorean's theorem:

`√(3)^2=1^2+b^2` and

`3=1+b^2` so

`b^2=2` and

`b=√(2)`

But since we are in QIV and that is a y value, it is negative.

Therefore,

`sin\theta=-(√(2) )/(√(3) )`

Rationalizing, you get

`sin\theta=-(√(6) )/(3)`