given that cos 5pi/9 = sin y first express 5pi/9 as a sum of pi/2 and a angle and then apply a confunction identity to determine the measure of angle y
Well, well, well, let's break this down, shall we?
We know that cos (5π/9) = sin y. But we're going to have to do some funny math to get there.
First, let's express 5π/9 as a sum of π/2 and another angle.
Well, 5π/9 is equal to π/2 + π/6. Why? Because let's face it, π/6 can make anything better.
So now we have cos (π/2 + π/6) = sin y.
Here comes the fun part – let's bust out a cosine function identity.
The identity we need to use is:
cos (a + b) = cos a cos b - sin a sin b.
In our case, a = π/2 and b = π/6.
So let's plug everything in:
cos (π/2) cos (π/6) - sin (π/2) sin (π/6) = sin y.
But we all know that cos (π/2) = 0 and sin (π/2) = 1, right?
So our equation simplifies to:
0 cos (π/6) - 1 sin (π/6) = sin y.
And you know what we get? -sin (π/6) = sin y.
So the measure of angle y is -π/6. Don't worry, y, it's not you, it's just some funny math!
To express 5π/9 as a sum of π/2 and an angle, we can start by finding the closest multiple of π/2 to 5π/9, which is 4π/9.
Now, we can rewrite 5π/9 as the sum of 4π/9 and π/9:
5π/9 = 4π/9 + π/9
Next, we can use the cofunction identity for cosine and sine to determine the measure of angle y.
The cofunction identity states that cos(θ) = sin(π/2 - θ).
Since cos(5π/9) = sin y, we can rewrite cos(5π/9) using the cofunction identity:
cos(5π/9) = sin(π/2 - y)
Therefore, we have:
sin(π/2 - y) = sin y
Now, we can equate the angles inside the sine function:
π/2 - y = y
Solving for y, we can add y to both sides:
π/2 - 2y = 0
Next, subtract π/2 from both sides:
-2y = -π/2
Finally, divide both sides by -2:
y = π/4
Therefore, the measure of angle y is π/4.
To express 5π/9 as a sum of π/2 and an angle, we first need to find the equivalent angle that lies between 0 and π/2. We can do this by subtracting multiples of π/2 from 5π/9 until we have an angle in the desired range.
Starting with 5π/9, we can subtract π/2:
5π/9 - π/2 = 10π/18 - 9π/18 = π/18
Now we have expressed 5π/9 as the sum of π/2 and π/18.
Next, let's apply a cofunction identity. The cofunction identity for sine and cosine states that the sine of an angle is equal to the cosine of its complementary angle, and the cosine of an angle is equal to the sine of its complementary angle.
Since we have cos(5π/9) = sin(y), we can rewrite it as sin(π/2 - 5π/9) = sin(y). Here, π/2 - 5π/9 is the complementary angle of 5π/9.
To determine the value of y, we can equate the angles:
π/2 - 5π/9 = y
Simplifying the equation:
9π/18 - 10π/18 = y
-y = -π/18
y = π/18
Therefore, the measure of angle y is π/18.
5/9 = 1/2 + 1/18
The co- in cosine means sine of the complement. That is
cos(x) = sin(pi/2 - x)
so, cos(5pi/9) = sin(pi/2 - 5pi/9)