Given that cos 13pi/18=sin y, first express 13pi/18 as a sum of pi/2 and an angle, and then apply a trigonometric identitiy to determine the measure of angle y.
13π/18 = 9π/18 + 4π/18 = π/2 + 2π/9
cos 13π/18 = -sin 2π/9
To express 13π/18 as a sum of π/2 and an angle, we need to find the nearest multiple of π/2 to 13π/18.
To obtain a common denominator, we can multiply both π and 18 by 2:
13π/18 = (13π * 2)/(18 * 2)
= 26π/36
Now, let's express 26π/36 as a sum of π/2 and another angle. We know that π/2 is equal to 18π/36, so subtracting 18π/36 from both sides gives:
26π/36 - 18π/36 = (26π - 18π)/36
= 8π/36
= 2π/9
Therefore, 13π/18 can be written as π/2 + 2π/9.
Now, let's apply a trigonometric identity to determine the measure of angle y.
Given: cos(13π/18) = sin(y)
Using the identity cos(θ) = sin(π/2 - θ), we can rewrite the equation:
sin(π/2 - θ) = sin(y)
Comparing the equation to the identity, we can see that π/2 - θ is equivalent to y.
Therefore, π/2 - θ = y.
Substituting the known value of θ as 2π/9, we have:
π/2 - 2π/9 = y
To simplify, we need to find a common denominator for π/2 and 2π/9:
π/2 = (π/2) * (9/9)
= (9π/18)
Substituting this into the equation, we have:
(9π/18) - (2π/9) = y
Now, let's simplify by obtaining a common denominator:
(9π - 4π)/18 = y
Combining like terms:
5π/18 = y
The measure of angle y is 5π/18.
To express 13π/18 as a sum of π/2 and an angle, we can simplify it using the fact that π/2 = 9π/18.
13π/18 = (9π/18) + (4π/18)
So, 13π/18 can be expressed as π/2 plus 4π/18.
Now, let's find the measure of angle y using the trigonometric identity.
Since cos(13π/18) = sin(y), we can use the identity sin(x) = cos(π/2 - x) to express sin(y) in terms of cos.
sin(y) = cos(π/2 - 13π/18)
Using the angle subtraction formula for cosine, we have:
sin(y) = cos(π/2)cos(13π/18) + sin(π/2)sin(13π/18)
Since cos(π/2) = 0 and sin(π/2) = 1, we can simplify it further:
sin(y) = 0 * cos(13π/18) + 1 * sin(13π/18)
sin(y) = sin(13π/18)
From this, we can conclude that angle y is equal to 13π/18.