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Sagot :
I got
[tex] - \frac{6 3}{65} [/tex]
What we know
cos a=-3/5.
sin b=12/13
Angle A interval are between 180 and 270 or third quadrant
Angle B quadrant is between 90 and 180 or second quadrant.
What we need to find
Cos(b)
Cos(a)
What we are going to apply
Sum and Difference Formulas
Basics Sine and Cosines Identies.
1. Let write out the cos(a-b) formula.
[tex] \cos(a - b) = \cos(a) \cos(b) + \sin(a) \sin(b) [/tex]
2. Use the interval it gave us.
According to the given, Angle B must between in second quadrant.
Since sin is opposite/hypotenuse and we are given a sin b=12/13. We. are going to set up an equation using the pythagorean theorem.
.
[tex] {12}^{2} + {y}^{2} = {13}^{2} [/tex]
[tex]144 + {y}^{2} = 169[/tex]
[tex]25 = {y}^{2} [/tex]
[tex]y = 5[/tex]
so our adjacent side is 5.
Cosine is adjacent/hypotenuse so our cos b=5/13.
Using the interval it gave us, Angle a must be in the third quadrant. Since cos is adjacent/hypotenuse and we are given cos a=-3/5. We are going to set up an equation using pythagorean theorem,
.
[tex]( - 3) {}^{2} + {x}^{2} = {5}^{2} [/tex]
[tex]9 + {x}^{2} = 25[/tex]
[tex] {x}^{2} = 16[/tex]
[tex]x = 4[/tex]
so our opposite side is 4. sin =Opposite/Hypotenuse so our sin a =4/5.Sin is negative in the third quadrant so
sin a =-4/5.
Now use cosine difference formula
[tex] - \frac{3}{5} \times \frac{5}{13} + - \frac {4}{5} \times \frac{12}{13} [/tex]
[tex] - \frac{15}{65} + ( - \frac{48}{65} )[/tex]
[tex] - \frac{63}{65} [/tex]
Hope this helps
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