If cos cos A=3/5 and sinB=7/25,where A is acute and B is obtuse,find without using tables the value of Cos(A+B)
To find the value of cos(A+B), we can use the cosine addition formula:
cos(A+B) = cos A * cos B - sin A * sin B
Given that cos cos A = 3/5, we can rewrite it as cos A = 3/5.
Also, sin B = 7/25.
Now, let's substitute these values into the cosine addition formula:
cos(A+B) = (3/5) * cos B - sin A * (7/25)
Since B is obtuse, we know that cos B is negative. Therefore, we can write cos B as -√(1 - sin^2 B).
Substituting sin B = 7/25 into cos B, we have:
cos B = -√(1 - (7/25)^2) = -√(1 - 49/625) = -√(576/625) = -24/25
Now, let's substitute the values into the formula:
cos(A+B) = (3/5) * (-24/25) - sin A * (7/25)
To find sin A, we can use the Pythagorean identity:
sin^2 A + cos^2 A = 1
Since cos A = 3/5, we can solve for sin A:
sin^2 A + (3/5)^2 = 1
sin^2 A = 1 - (9/25)
sin^2 A = 16/25
sin A = 4/5 (since sin A is positive)
Now we can substitute sin A = 4/5 and cos B = -24/25 into the formula:
cos(A+B) = (3/5) * (-24/25) - (4/5) * (7/25)
cos(A+B) = -72/125 - 28/125
cos(A+B) = -100/125
cos(A+B) = -4/5
Therefore, the value of cos(A+B) is -4/5 without using tables.
To find the value of cos(A + B) without using tables, we can use the cosine sum identity:
cos(A + B) = cos A * cos B - sin A * sin B
Given:
cos(cos A) = 3/5
sin B = 7/25
To solve the problem, we need to find the values of cos A and sin A.
We know that cos^2 A + sin^2 A = 1 from the Pythagorean identity. Rearranging the equation, we get:
sin^2 A = 1 - cos^2 A
Using the given value, substitute cos A = 3/5 into the equation:
sin^2 A = 1 - (3/5)^2
sin^2 A = 1 - 9/25
sin^2 A = 16/25
sin A = square root of (16/25)
sin A = 4/5 (since A is an acute angle, sin A is positive)
Now we have cos A = 3/5 and sin A = 4/5.
Substituting these values along with sin B = 7/25 into the cosine sum identity, we can calculate cos(A + B):
cos(A + B) = (3/5)*(7/25) - (4/5)*(7/25)
cos(A + B) = 21/125 - 28/125
cos(A + B) = -7/125
Therefore, the value of cos(A + B) is -7/125, where A is acute and B is obtuse.
but cos(A+B) = cos A cos B - sin A sin B
for angle A 3^2+4^2 = 5^2
cos A = 3/5 so sin A = 4/5
for angle B 24^2 + 7^2 = 25^2
cos B = -24/25 and sin B = 7/25 because B is obtuse