From the given figure
Triangle ABC is the right angle triangle where C is the 90 degree
1)
Here we have θ=28, we need to calculate the Side of AB
Since AB is the hypotenuse of the triangle ABC and the side BC = 3ft, which is the base of the triangle ABC
Thus, apply the trignometric ratio of Cos
[tex]\begin{gathered} \text{Cos}\theta=\frac{Base}{Hypotenuse} \\ \text{Cos (28)=}\frac{BC}{AB} \\ \text{Cos (28)=}\frac{3}{AB} \\ AB=\frac{3}{Cos(28)} \\ AB=\frac{3}{0.882} \\ AB=3.401\text{ f}eet \end{gathered}[/tex]
Answer : AB = 3.041 feet
2)
Here we have the string is stretched upto 5.2 feet i.e AB = 5.2 feet
We need to find the value of the angle,
thus again apply the trignometric ratio of Cos
[tex]\begin{gathered} \text{Cos}\theta=\frac{Base}{Hypotenuse} \\ \text{Cos (}\theta\text{)=}\frac{BC}{AB} \\ \text{Cos (}\theta\text{)=}\frac{3}{5.2} \\ \text{Cos(}\theta)=0.576 \\ \theta=\cos ^{-1}(0.576) \\ \theta=54.83^{\circ} \end{gathered}[/tex]
So, the angle θ = 54.83
Answer : θ = 54.83
3)
when the θ=15 then the AB will be :
[tex]\begin{gathered} \text{Cos}\theta=\frac{Base}{Hypotenuse} \\ \text{Cos (15)=}\frac{BC}{AB} \\ \text{Cos (15)=}\frac{3}{AB} \\ AB=\frac{3}{\text{Cos}(15)} \\ AB=\frac{3}{0.759} \\ AB=3.952 \end{gathered}[/tex]
thus, at angle 15 the side AB = 3.1059 feet
Now, at an angle 30 the AB will be :
[tex]\begin{gathered} \text{Cos}\theta=\frac{Base}{Hypotenuse} \\ \text{Cos (30)=}\frac{BC}{AB} \\ \text{Cos (30)=}\frac{3}{AB} \\ AB=\frac{3}{\text{Cos}(30)} \\ AB=\frac{3}{0.866} \\ AB=3.464\text{ f}eet \end{gathered}[/tex]
So, the collar will slide from point A to B when the angle changes from 15 to 30 is the difference between the side AB at 15 to AB at 30
Difference = 3.952 -3.464
Difference = 0.488 feet
Answer : 0.488 feet
