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
come on, man. This is 4th grade addition of fractions.
13/18 = 9/18 + 5/18
now, cos(13π/18) = cos(π/2) cos(5π/18) - sin(π/2) sin(5π/18) = -sin(5π/18)
so y = -5π/18
If you want a positive angle, just add 2π to that.
13 π / 18 = π / 2 + θ
Subtract π / 2 to both sides
13 π / 18 - π / 2 = θ
13 π / 18 - 9 π / 18 = θ
4 π / 18 = θ
2 ∙ 2 π / 2 ∙ 9 = θ
2 π / 9 = θ
θ = 2 π / 9
13 π / 18 = π / 2 + 2 π / 9
Use identity:
cos ( π / 2 + θ ) = - sin ( θ )
cos ( 13 π / 18 ) = cos ( π / 2 + 2 π / 9 ) = - sin ( 2 π / 9 )
Now:
cos ( 13 π / 18 ) = sin ( y )
- sin ( 2 π / 9 ) = sin ( y )
sin ( y ) = - sin ( 2 π / 9 )
The solutions are:
y = - 2 π / 9 + 2 π n
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Since:
sin ( π + θ ) = - sin( θ )
sin ( π + 2 π / 9 ) = - sin ( 2 π / 9 )
sin ( 9 π / 9 + 2 π / 9 ) = - sin ( 2 π / 9 )
sin ( 11 π / 9 ) = - sin ( 2 π / 9 )
y = 11 π / 9 + 2 π n
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oops. How ever did I get 5/18?
Luckily, it was Bosnian for the save!
To express 13π/18 as a sum of π/2 and an angle, we need to find a common denominator for 13 and 18, which is 36.
13π/18 = 13π * 2/18 * 2 = (26π/36) = (13π/36) * 2
Now, let's find a trigonometric identity to determine the measure of angle y.
Since cos(θ) = sin(π/2 - θ), we can rewrite cosine in terms of sine:
cos(13π/18) = sin(π/2 - y)
Comparing this with the given equation cos(13π/18) = sin(y), we can get:
sin(y) = sin(π/2 - y)
Now, using the identity sin(π - x) = sin(x), we can rewrite:
sin(π/2 - y) = sin(y)
This implies that π/2 - y = y
Solving for y:
π/2 = 2y
y = π/4
Therefore, the measure of angle y is π/4.