Given that for positive acute angles A and B:
[tex]\begin{gathered} \cos A=\frac{60}{61} \\ \\ \tan B=\frac{12}{35} \end{gathered}[/tex]
You need to remember that, by definition:
[tex]\begin{gathered} \cos \alpha=\frac{adjacent}{hypotenuse} \\ \\ \tan \alpha=\frac{opposite}{adjacent} \end{gathered}[/tex]
Look at the Right Triangle shown below:
It is also important to remember the following Trigonometric Identity:
[tex]\cos \mleft(A-B\mright)=cosA\cdot cosB+sinA\cdot sinB[/tex]
Then, in order to solve this exercise, you need to find:
[tex]\begin{gathered} \cos B \\ \sin A \\ \sin B \end{gathered}[/tex]
Using the data given in the exercise, you can draw these two Right Triangles (they are not drawn to scale):
Since by definition:
[tex]\sin \alpha=\frac{opposite}{hypotenuse}[/tex]
You need to find the opposite side of the first triangle and the hypotenuse of the second triangle. You can do this by applying the Pythagorean Theorem, which states that:
[tex]c^2=a^2+b^2[/tex]
Where "c" is the hypotenuse, and "a" and "b" are the legs of the Right Triangle.
Then, the opposite side of the first triangle is:
[tex]\begin{gathered} 61^2=60^2+b^2 \\ \\ 61^2-60^2=b^2 \\ \\ \sqrt[]{61^2-60^2}=b \\ \\ b=11 \end{gathered}[/tex]
And the hypotenuse of the second triangle is:
[tex]\begin{gathered} c^2=12^2+35^2 \\ \\ c=\sqrt[]{12^2+35^2} \\ \\ c=37 \end{gathered}[/tex]
Now you can determine that:
[tex]\begin{gathered} \cos B=\frac{35}{37} \\ \\ \sin A=\frac{11}{61} \\ \\ \sin B=\frac{12}{37} \end{gathered}[/tex]
Substitute values into the Trigonometric Identity:
[tex]\begin{gathered} \cos (A-B)=cosA\cdot cosB+sinA\cdot sinB \\ \\ \cos (A-B)=(\frac{60}{61})(\frac{35}{37})+(\frac{11}{61})(\frac{12}{37}) \end{gathered}[/tex]
Simplify:
- Multiply the fractions:
[tex]\begin{gathered} \cos (A-B)=\frac{60\cdot35}{61\cdot37}+\frac{11\cdot12}{61\cdot37} \\ \\ \cos (A-B)=\frac{2100}{2257}+\frac{132}{2257} \end{gathered}[/tex]
- Add the fractions:
[tex]\begin{gathered} \cos (A-B)=\frac{2100+132}{2257} \\ \\ \cos (A-B)=\frac{2232}{2257} \end{gathered}[/tex]
Hence, the answer is:
[tex]\cos (A-B)=\frac{2232}{2257}[/tex]
