Find cos(s+t) if cos s= -1/2 and sin t= 3/5, s and t are in quadrant II.
I got the answer to be -4/10 plus -3sqrt(3)/10. Is that right?
Then a second problem was find sin (s+t) if sin s=2/3 and sin t=-1/3, sin is in quadrant II and t in quadrant IV. I got answer of 2sqrt(8/9) plus -sqrt(5)/9.
Can someone tell me if I did these right because I have a test tomorrow tahnks!
the first one is correct, for the second I got
(2√8 + √5)/9
or
(4√2 + √5)/9
To find cos(s+t) and sin(s+t), we can use the trigonometric identities involving sum of angles. Here's how you can solve each problem step by step:
Problem 1: Find cos(s+t) with given values cos s = -1/2 and sin t = 3/5, where s and t are in quadrant II.
To find cos(s+t), we can use the formula:
cos(s+t) = cos s * cos t - sin s * sin t
1. First, recall that in quadrant II, cos x is negative and sin x is positive. Therefore, cos s = -1/2 means that the adjacent side is negative and the hypotenuse is positive, while sin t = 3/5 means that the opposite side is positive while the hypotenuse is positive.
2. To find cos t, we can use the Pythagorean identity:
sin^2 t + cos^2 t = 1
Substituting sin t = 3/5, we have:
(3/5)^2 + cos^2 t = 1
9/25 + cos^2 t = 1
cos^2 t = 1 - 9/25
cos^2 t = 16/25
cos t = ± √(16/25)
cos t = ± 4/5
Since t is in quadrant II, cos t is negative. Therefore, cos t = -4/5.
3. Now we can calculate cos(s+t) using the formula:
cos(s+t) = cos s * cos t - sin s * sin t
Substituting the given values, we have:
cos(s+t) = (-1/2) * (-4/5) - sin s * sin t
cos(s+t) = 2/5 - (-1/2) * (3/5)
cos(s+t) = 2/5 + 3/10
cos(s+t) = 4/10 + 3/10
cos(s+t) = 7/10
Therefore, the correct answer is cos(s+t) = 7/10.
Problem 2: Find sin(s+t) with given values sin s = 2/3 and sin t = -1/3, where sin s is in quadrant II and sin t is in quadrant IV.
To find sin(s+t), we can again use the formula:
sin(s+t) = sin s * cos t + cos s * sin t
1. In quadrant II, sin x is positive. Therefore, sin s = 2/3 means the opposite side is positive while the hypotenuse is positive.
2. In quadrant IV, sin x is negative and cos x is positive. Therefore, sin t = -1/3 means the opposite side is negative, while the hypotenuse is positive.
3. Now we can calculate sin(s+t) using the formula:
sin(s+t) = sin s * cos t + cos s * sin t
Substituting the given values, we have:
sin(s+t) = (2/3) * cos t + cos s * (-1/3)
sin(s+t) = 2/3 * cos t - cos s/3
Since we don't have enough information about the value of cos t or cos s, we cannot simplify the expression further without more information.
Therefore, we cannot determine the exact value of sin(s+t) using the given information.
Please note that the answer depends on the exact values of cos t and cos s, which we don't have in this case.