What are the values of the trigonometric ratios for this triangle? Match the correct ratio to each item.

We know in a right triangle we have three major side lengths. One of these side lengths must be larger than the other two. This is the hypotenuse. To help solve this problem, let's define what sine, cosine, and tangent are in terms of side lengths.

#sin(theta) = "opposite"/"hypotenuse"#
#cos(theta) = "adjacent"/"hypotenuse"#
#tan(theta) = "opposite"/"adjacent"#

Out of all the options we have, the largest number is 13, meaning the largest side length, or the length of the hypotenuse, must be 13. The only ratio above that does not use the hypotenuse is tangent, so we know that the number ratio that does not have 13 in it is for tangent.

This allows us to pair #tan(theta)# with number 2 (5/12).

Now that we know #tan(theta) = 5/12# We can accurately determine adjacent and opposite side lengths based on the formula for tangent, seen above.

From that we determine that opposite is 5, while adjacent is 12. From here it's a matter of plugging in and matching.

#sin(theta) = "opposite"/"hypotenuse"#, so #sin(theta)=5/13# or option 3.

#cos(theta) = "adjacent"/"hypotenuse"#, so #cos(theta) = 12/13# or option 1.