Find sin 𝜃 given 51 cos 90 degrees minus 𝜃 is equal to 24, where 𝜃 is a positive acute angle.
This is a question about cofunction identities, and one in particular that’s cos of 90 degrees minus 𝜃 is equal to sin 𝜃. You can see that this identity is true by taking a right triangle.
If this angle has measure 𝜃, then as the sum of the measures of the angles in a triangle is equal to 180 degrees, the other angle must be 90 degrees minus 𝜃.
Sin 𝜃 is equal to the length of the side opposite the angle, divided by the length of the hypotenuse.
Okay and what about cos of 90 degrees minus 𝜃? Well that’s equal to the length of the side adjacent to that angle, divided by the length of the hypotenuse. And so using this right triangle, we can see that sin 𝜃 and cos 90 degrees minus 𝜃 are the same ratio of sides, and so they are equal.
This means that we can replace cos of 90 degrees minus 𝜃 by sin 𝜃. So we get 51 sin 𝜃 is equal to 24. Dividing both sides by 51, we get that sin 𝜃 is equal to 24 over 51. And we can simplify our fraction by dividing both numerator and denominator by three to get that sin 𝜃 is equal to eight over 17.