If cot(x) = 5/12, what is sin(x)?

If cot(x) = 5/12, what is sin(x)?

Step 1: Understand that cot(x) is the ratio of cos(x) to sin(x). So, cot(x) = cos(x) / sin(x).
Step 2: Given cot(x) = 5/12, we can write this as cos(x) / sin(x) = 5/12.
Step 3: Let sin(x) = 12k and cos(x) = 5k for some value k. This keeps the ratio the same.
Step 4: Use the Pythagorean theorem, which states that sin^2(x) + cos^2(x) = 1.
Step 5: Substitute sin(x) and cos(x) into the equation: (12k)^2 + (5k)^2 = 1.
Step 6: Simplify the equation: 144k^2 + 25k^2 = 1, which gives 169k^2 = 1.
Step 7: Solve for k: k^2 = 1/169, so k = 1/13.
Step 8: Now find sin(x): sin(x) = 12k = 12 * (1/13) = 12/13.

Trigonometric Identities – Understanding the relationship between cotangent, sine, and cosine using the identity cot(x) = cos(x)/sin(x).
Pythagorean Theorem in Trigonometry – Applying the Pythagorean theorem to find the lengths of the sides of a right triangle when given one of the trigonometric ratios.