Answer:
39/46
Step-by-step explanation:
Now, the key to answer this first is knowing the value of cos θ
Mathematically, when we have sin θ
What we have is the ratio of the opposite to the hypotenuse side
Thus, here, since sin θ = 5/13, this means that the opposite is 5 while the hypotenuse is 13
Now to complete the 3rd side of the triangle, we need to use the Pythagoras’s theorem
This states that the square of the length of the hypotenuse equals the sum of the squares of the two other sides
So let’s say the adjacent or the third side is d
This means that;
13^2 = 5^2 + d^2
d^2 = 13^2 - 5^2
d^2 = 169-25
d^2 = 144
d = √(144)
d = 12
The cosine of the angle mathematically is the ratio of length of the adjacent to that of the hypotenuse
and that is 12/13
Hence Cos θ = 12/13
What we need last to answer the question is cos2 θ
Using trigonometric identity;
Cos2θ = cos^2 θ - sin^2 θ
Inputing the values of sine and cos of the angle theta, we have;
cos2θ = (12/13)^2 - (5/13)^2
cos2θ = 144/169 - 25/169 = 119/169
Thus;
cosθ/(cos2θ + sinθ) = 12/13/(119/169 + 5/13)
= 12/13/(184/169)
= 12/13÷ 184/169
= 12/13 * 169/184
= (13 * 3)/46 = 39/46
