Write the expression in terms of the appropriate cofunction.cos(19π45)\cos\left(\frac{19\pi}{45}\right)cos(4519π)

sin ⁡ ( 7 π 90 ) \sin\left(\frac{7\pi}{90}\right) sin ( 90 7 π )

Write the expression in terms of the appropriate cofunction. cos ( 19 π 45 ) \cos\left(\frac{19\pi}{45}\right) cos ( 45 19 π )

Let's give this problem a try. So here we are asked to write the expression in terms of the appropriate cofunction, and we're given the cosine of 19pi over 45. Now to solve this problem, our first step should be to find the cofunction of cosine, and it turns out that that's just going to be sine. We've learned that in previous videos. Now from here what we wanna do is take the sine of the complementary angle, and the complementary angle to 19 pi over 45 is just going to be pi over 2 minus the angle that we started with, which is 19 pi over 45. So this is going to be the complementary angle. Now from here, what we need to do is combine these fractions. And to do this, well, there's not a real easy or straightforward way, but what I can do is multiply the top and bottom of this fraction on the left by 45, and then multiply the top and bottom of this fraction on the right by 2, because this is going to allow me to get common denominators. Right? So what we're going to end up with is the sign of pi over 2 times 45 over 45, which would be 45 pi on the top, and then 45 times 2 is 90, and this is going to be minus 19 pi times 2. So we would have 19 times 2 which is 38 pi, and that we have this pi still here. And then this is divided by 45 times 2 which is also 90. And notice how we get like denominators, so we can combine the tops of the fractions. So what we're going to have is the sine, and keep in mind this is all still within the sine operation, and then we have 45 minus 38. Well, 45 minus 38 is 7, so we're going to have 7 pi. We have to keep this pi right here, and this is all going to be divided by the common denominator which is 90. And there's no real way we can simplify from here, so this is going to be our final answer for this cofunction. So that is how we can find the appropriate cofunction for the expression we started with. Hope you found this video helpful. Thanks for watching.

Write the expression in terms of the appropriate cofunction. cos ⁡ ( 19 π 45 ) \cos\left(\frac{19\pi}{45}\right) cos ( 45 19 π )

sin ⁡ ( 7 π 90 ) \sin\left(\frac{7\pi}{90}\right) sin ( 90 7 π )

cos ⁡ ( 7 π 90 ) \cos\left(\frac{7\pi}{90}\right) cos ( 90 7 π )

sin ⁡ ( 4031 π 45 ) \sin\left(\frac{4031\pi}{45}\right) sin ( 45 4031 π )

cos ⁡ ( 4031 π 90 ) \cos\left(\frac{4031\pi}{90}\right) cos ( 90 4031 π )

2. Trigonometric Functions on Right Triangles

Cofunctions of Complementary Angles

Trigonometry

2. Trigonometric Functions on Right Triangles

Cofunctions of Complementary Angles

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Multiple Choice

Write the expression in terms of the appropriate cofunction.
$\cos\left(\frac{19\pi}{45}\right)$

$\cos\left(\frac{7\pi}{90}\right)$

$\sin\left(\frac{4031\pi}{45}\right)$

$\sin\left(\frac{7\pi}{90}\right)$

$\cos\left(\frac{4031\pi}{90}\right)$

Verified step by step guidance

Identify the cofunction identities: The cofunction identities relate sine and cosine functions. Specifically, $ \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) $ and $ \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) $.

Convert the angle in the cosine expression $ \cos\left(\frac{19\pi}{45}\right) $ to its cofunction: Use the identity $ \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) $. Calculate $ \frac{\pi}{2} - \frac{19\pi}{45} $.

Simplify the expression $ \frac{\pi}{2} - \frac{19\pi}{45} $: Find a common denominator to combine the fractions, which is 90. This gives $ \frac{45\pi}{90} - \frac{38\pi}{90} = \frac{7\pi}{90} $.

Apply the cofunction identity to the sine expression $ \sin\left(\frac{4031\pi}{45}\right) $: Use the identity $ \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) $. Calculate $ \frac{\pi}{2} - \frac{4031\pi}{45} $.

Simplify the expression $ \frac{\pi}{2} - \frac{4031\pi}{45} $: Again, find a common denominator, which is 90. This gives $ \frac{45\pi}{90} - \frac{8062\pi}{90} = -\frac{8017\pi}{90} $. Since cosine is an even function, $ \cos(-\theta) = \cos(\theta) $, so the expression becomes $ \cos\left(\frac{4031\pi}{90}\right) $.

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