Answer:
[tex]\cos (A -B)=\dfrac{2232}{2257}[/tex]
Step-by-step explanation:
To find the exact value of cos(A - B), we can use the angle subtraction formula for cosine:
[tex]\boxed{\cos (A -B)=\cos A \cos B + \sin A \sin B}[/tex]
First, find the value of sin A and cos A.
The tangent of an angle in a right triangle is defined as the ratio of the side opposite the angle to the side adjacent to the angle. Given that tan A = 11/60, this implies that the side opposite angle A measures 11 units, and the side adjacent to angle A measures 60 units.
To determine the length of the hypotenuse H, we can apply the Pythagorean Theorem, which states that the square of the hypotenuse of a right triangle equals the sum of the squares of its other two sides:
[tex]H^2=11^2+60^2 \\\\H^2=121+3600 \\\\ H^2=3721 \\\\ H=\sqrt{3721}\\\\H=61[/tex]
The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Similarly, the cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Therefore:
[tex]\sin A= \dfrac{11}{61} \\\\\\ \cos A =\dfrac{60}{61}[/tex]
Now, find cos B by using the Pythagorean identity, sin²θ + cos²θ = 1:
[tex]\sin^2 B + \cos^2 B = 1 \\\\\\\left(\dfrac{12}{37}\right)^2+ \cos^2 B = 1 \\\\\\\cos^2 B=1-\left(\dfrac{12}{37}\right)^2 \\\\\\\cos^2 B=\dfrac{1225}{1369} \\\\\\\cos B=\pm \sqrt{\dfrac{1225}{1369}} \\\\\\\cos B=\pm \dfrac{35}{37}[/tex]
Given that angle B is a positive acute angle, it implies that B lies in the first quadrant, where both sine and cosine are positive. Therefore, we take the positive square root:
[tex]\cos B = \dfrac{35}{37}[/tex]
Finally, substitute the values of cos A, cos B, sin A and sin B into the angle subtraction formula for cosine:
[tex]\cos (A -B)=\dfrac{60}{61} \cdot \dfrac{35}{37} + \dfrac{11}{61} \cdot\dfrac{12}{37} \\\\\\ \cos (A -B)=\dfrac{2100}{2257} + \dfrac{132}{2257} \\\\\\ \cos (A -B)=\dfrac{2232}{2257} \\\\\\[/tex]
Therefore, the value of cos(A - B) in simplest form is:
[tex]\Large\text{$\cos (A -B)=\boxed{\dfrac{2232}{2257}}$}[/tex]
