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.