Angle a lies in the second quadrant and angle b lies in the third quadrant such that cos(a)=-3/5 and tan(b)=24/7. Determine an exact value for
a) cos(a+b)
b) sin(a-b)
c) sin2a
question ( c)
please atemt question (c)
Yes
To determine the exact values for those expressions, we will need to use trigonometric identities and properties.
a) cos(a + b)
To find the exact value of cos(a + b), we can use the cosine angle addition identity:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
Given that cos(a) = -3/5 and tan(b) = 24/7, we can find sin(a) and cos(b) using the Pythagorean identity.
Since angle a lies in the second quadrant, cos(a) is negative and sin(a) will be positive. We can use the Pythagorean identity for sine and cosine:
cos^2(a) + sin^2(a) = 1
Substituting the value of cos(a) = -3/5:
(-3/5)^2 + sin^2(a) = 1
9/25 + sin^2(a) = 1
sin^2(a) = 1 - 9/25
sin^2(a) = 16/25
sin(a) = √(16/25)
sin(a) = 4/5
Now, let's find cos(b) using the given tangent value:
tan(b) = sin(b) / cos(b) = 24/7
sin(b) = 24
cos(b) = 7
Note: We took sin(b) = 24 and cos(b) = 7 since angle b lies in the third quadrant where sin(b) is negative.
Now, substitute the values into the cosine angle addition identity:
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
cos(a + b) = (-3/5)(7) - (4/5)(24)
cos(a + b) = -21/5 - 96/5
cos(a + b) = -117/5
Therefore, cos(a + b) = -117/5.
b) sin(a - b)
To find the exact value of sin(a - b), we can use the sine angle subtraction identity:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
Now we know sin(a) = 4/5, cos(b) = 7, and sin(b) = 24 (from the previous calculations).
Substituting the values into the sine angle subtraction identity:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
sin(a - b) = (4/5)(7) - (-3/5)(24)
sin(a - b) = 28/5 + 72/5
sin(a - b) = 100/5
Therefore, sin(a - b) = 20.
c) 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)
From the previous calculations, we know sin(a) = 4/5 and cos(a) = -3/5.
Substituting the values into the double-angle formula for sine:
sin(2a) = 2sin(a)cos(a)
sin(2a) = 2(4/5)(-3/5)
sin(2a) = -24/25
Therefore, sin(2a) = -24/25.
a) cos(a)cos(b) + sin(a)sin(b)
find sin ratio: use pythagorus (5^2) - (-3^2) = #. Sin = y/r = #/r.
then solve a)