If tan(x) = 3/4, what is the value of sin(x)?

If tan(x) = 3/4, what is the value of sin(x)?

Correct Answer: 3/5

Step 1: Understand that tan(x) = sin(x) / cos(x). We are given that tan(x) = 3/4.
Step 2: This means that sin(x) = 3k and cos(x) = 4k for some value k.
Step 3: Use the Pythagorean theorem to find the hypotenuse. The formula is sin^2(x) + cos^2(x) = 1.
Step 4: Substitute sin(x) and cos(x) into the equation: (3k)^2 + (4k)^2 = 1.
Step 5: Simplify the equation: 9k^2 + 16k^2 = 1, which gives 25k^2 = 1.
Step 6: Solve for k: k^2 = 1/25, so k = 1/5.
Step 7: Now find sin(x): sin(x) = 3k = 3 * (1/5) = 3/5.

Trigonometric Identities – Understanding the relationship between sine, cosine, and tangent functions.
Pythagorean Theorem – Using the theorem to find the hypotenuse in a right triangle given the lengths of the opposite and adjacent sides.