If x and y are complementary angles and 5 tan x-12=0,calculate by using a diagram the value of 2sin(squared)x+cos y.

sorry, no diagrams here

from 5tanx - 12 = 0

tanx = 12/5

make a sketch of a right-angled triangle, in quad I
where x is the angle at the origin, the other acute angle is y
the hypotenuse would be 13 , from the 5-12-13 right-angled triangle

then 2sin^2 x + cosy
= 2(12^2 / 13^2) + 12/13
= 288/169 + 12/13
= 444/169

To find the value of 2sin^2(x) + cos(y) using a diagram, we first need to solve for the values of x and y.

Given that x and y are complementary angles, we know that the sum of their measures is 90 degrees.

From the equation 5tan(x) - 12 = 0, we can solve for x as follows:

5tan(x) - 12 = 0
5tan(x) = 12
tan(x) = 12/5

Using the inverse tangent (tan^(-1)) function, we can find the value of x:

x = tan^(-1)(12/5)

Now that we have the value of x, we can find the measure of y by subtracting x from 90 degrees:

y = 90 - x

Next, let's calculate the value of 2sin^2(x) + cos(y) using the given values of x and y.

Step 1: Substitute the values of x and y:

2sin^2(x) + cos(y) = 2sin^2(tan^(-1)(12/5)) + cos(90 - tan^(-1)(12/5))

Step 2: Evaluate sin^2(x) using the identity sin^2(x) = 1 - cos^2(x):

2(1 - cos^2(tan^(-1)(12/5))) + cos(90 - tan^(-1)(12/5))

Step 3: Simplify the expression:

2 - 2cos^2(tan^(-1)(12/5)) + cos(90 - tan^(-1)(12/5))

Step 4: Use the identity cos(90 - θ) = sin(θ):

2 - 2cos^2(tan^(-1)(12/5)) + sin(tan^(-1)(12/5))

Step 5: Use the fact that sin(θ) = cos(90 - θ):

2 - 2cos^2(tan^(-1)(12/5)) + cos^2(tan^(-1)(12/5))

Step 6: Combine like terms:

2 - cos^2(tan^(-1)(12/5))

Now, you can use a calculator to find the numerical value of cos^2(tan^(-1)(12/5)) and then substitute it back into the expression to get the final result.