How do you solve:
sin[arccos(-2/7)] ?
cos is - in quadrants 2 and 3
adjacent is -2, hypotenuse is 7, what is opposite?
7^2 - 2^2 = 49 - 4 = 45 = 9 * 5
sqrt(9*5) = 3 sqrt (5)
so sin = +/- 3 sqrt(5) /7
Thank you!
To solve sin[arccos(-2/7)], we can use the identity sin(arccos(x)) = sqrt(1 - x^2).
First, let's find the value of arccos(-2/7):
cos(arccos(-2/7)) = -2/7
Since the cosine function is the inverse of the arccosine function, we can write:
arccos(-2/7) = cos^(-1)(-2/7)
To find the value of cos^(-1)(-2/7), we can use a calculator or a table of trigonometric values.
Let's assume it is approximately equal to 1.928 radians.
Now we can substitute this value into the identity:
sin[arccos(-2/7)] = sin(1.928)
Using a calculator, we find that sin(1.928) is approximately equal to 0.935.
Therefore, sin[arccos(-2/7)] ≈ 0.935.
To solve sin(arccos(-2/7)), we need to use the relationship between sine and cosine in a right triangle.
Let's consider an arbitrary right triangle with an angle θ. The cosine of θ is defined as the ratio of the adjacent side to the hypotenuse, which is represented as cos(θ) = adjacent side / hypotenuse.
In this case, we have arccos(-2/7) as our angle θ. So, we need to find the value of the adjacent side and the hypotenuse in order to calculate cos(θ).
To find the adjacent side, we can assign a variable to it, let's say x. Therefore, x = -2.
To find the hypotenuse, we can assign a variable to it as well, let's say y. Therefore, y = 7.
Now, we can find the sine of the angle using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In equation form, this is written as: y^2 = x^2 + adjacent side^2.
Substituting the values we found, we get: 7^2 = -2^2 + adjacent side^2.
Simplifying further: 49 = 4 + adjacent side^2.
Rearranging the equation: adjacent side^2 = 49 - 4.
adjacent side^2 = 45.
Finally, we can solve for the adjacent side by taking the square root of both sides: adjacent side = √45.
Now that we have the values for the adjacent side and hypotenuse, we can calculate the sine of the angle using sin(θ) = opposite side / hypotenuse.
Since we know that the opposite side and hypotenuse are perpendicular to each other, the opposite side in this case is the square root of 45.
Therefore, we have sin(arccos(-2/7)) = √45 / 7.
In summary, sin(arccos(-2/7)) is equal to the square root of 45 divided by 7.