If cot(a)=3 and sin(b)= (15/17) where a being alpha lies in quadrant 3 and b being beta lies in quadrant 2, what is the exact value of sin(2a)
To find the exact value of sin(2a), we can use the double angle formula for sine:
sin(2a) = 2sin(a)cos(a)
We are given that cot(a) = 3, which is the reciprocal of tangent (tan(a)). Since tangent is negative in the third quadrant, we can determine that tan(a) = -1/3.
Using the Pythagorean identity for tangent:
tan^2(a) + 1 = sec^2(a)
(-1/3)^2 + 1 = sec^2(a)
1/9 + 1 = sec^2(a)
10/9 = sec^2(a)
Taking the square root of both sides:
√(10/9) = sec(a)
Since secant is the reciprocal of cosine, we have:
sec(a) = 3/√10
Now, we can substitute the values into the double-angle formula for sine:
sin(2a) = 2sin(a)cos(a)
= 2(sin(a) * 1/√10) * (3/√10)
= 6sin(a)/10
Since sin(a) is not given directly, we need to find it. We are given that sin(b) = 15/17, where b lies in the second quadrant. Using the Pythagorean theorem:
sin^2(b) + cos^2(b) = 1
(15/17)^2 + cos^2(b) = 1
225/289 + cos^2(b) = 1
cos^2(b) = 289/289 - 225/289
cos^2(b) = 64/289
Taking the square root of both sides:
cos(b) = ±√(64/289)
cos(b) = ±8/17
Since b lies in the second quadrant, cosine is positive, so cos(b) = 8/17.
Using the Pythagorean identity for sine:
sin^2(b) + cos^2(b) = 1
(15/17)^2 + (8/17)^2 = 1
225/289 + 64/289 = 1
289/289 = 1
This verifies that sin(b) = 15/17 and cos(b) = 8/17.
Now, we can find sin(a) using the following relation between the sine and cosine of complementary angles:
sin(a) = cos(b)
Therefore, sin(a) = 8/17.
Substituting this value into the expression for sin(2a):
sin(2a) = 6sin(a)/10
= 6(8/17)/10
= 48/170
= 24/85
Therefore, the exact value of sin(2a) is 24/85.
To find the exact value of sin(2a), we will use the double-angle formula for sine. The double-angle formula states that sin(2a) = 2 * sin(a) * cos(a).
Given that cot(a) = 3, we can use the relationship between cotangent and sine and cosine. Cotangent is the reciprocal of tangent, and tangent is equal to sine divided by cosine. Therefore, we have cot(a) = 1/tan(a) = 1/(sin(a)/cos(a)) = cos(a)/sin(a).
Since cot(a) = 3, we can write this as cos(a)/sin(a) = 3. Rearranging the equation, we get cos(a) = 3sin(a).
Now, we need to find the value of sin(a). Since a lies in quadrant 3, we know that the sine of a is negative. However, we don't have the exact value of sin(a). So, we need to find sin(a) using the given information.
Given that sin(b) = 15/17 and b lies in quadrant 2, we can use the Pythagorean identity for sine, which states that sin^2(b) + cos^2(b) = 1.
Substituting sin(b) = 15/17, we have (15/17)^2 + cos^2(b) = 1. Simplifying this equation, we get 225/289 + cos^2(b) = 1. Rearranging, we find cos^2(b) = 1 - 225/289. Solving this equation gives cos^2(b) = 64/289.
Since b lies in quadrant 2, cosine is negative. Therefore, we take the negative square root of cos^2(b). Hence, cos(b) = -8/17.
Now that we have cos(b) = -8/17, we can use the fact that cos(a) = 3sin(a) to find sin(a). Substituting cos(a) = 3sin(a), we get -8/17 = 3sin(a). Rearranging this equation, we find sin(a) = -8/51.
Finally, we can substitute the values of sin(a) and cos(a) into the double angle formula for sine: sin(2a) = 2 * sin(a) * cos(a).
Substituting sin(a) = -8/51 and cos(a) = 3sin(a), we get sin(2a) = 2 * (-8/51) * (3 * (-8/51)).
Simplifying this equation gives sin(2a) = -384/867.
Therefore, the exact value of sin(2a) is -384/867.