In a right triangle, angle θ sits at the bottom-left. The horizontal leg (along the bottom) has length 4 and the vertical leg has length 3. What is sin(θ)?
A4/5
B3/5
C3/4
D4/3
sin(θ) = opposite/hypotenuse. The side *opposite* angle θ (at bottom-left) is the vertical leg (length 3). The hypotenuse = √(3² + 4²) = 5. So sin(θ) = 3/5. The answer 4/5 is cos(θ) — using the adjacent leg instead of the opposite. Identifying which side is opposite is always relative to the angle in question.
Question 2 True / False
If nearly every side of a right triangle is doubled, the value of sin(θ) for any given angle θ in the triangle also doubles.
TTrue
FFalse
Answer: False
Trig ratios are ratios of sides, so scaling all sides by the same factor cancels out. If opposite = 3 and hypotenuse = 5, then sin = 3/5. After doubling: opposite = 6, hypotenuse = 10, sin = 6/10 = 3/5 — unchanged. The angle determines the ratio; the size of the triangle does not.
Question 3 Short Answer
Given sin(θ) = 3/5, what is csc(θ), and what is the general relationship between a trig function and its reciprocal counterpart?
Think about your answer, then reveal below.
Model answer: csc(θ) = 5/3. Each reciprocal function is the multiplicative inverse of its primary function: csc = 1/sin, sec = 1/cos, cot = 1/tan.
The three reciprocal functions (cosecant, secant, cotangent) do not introduce new information — they are algebraic inverses of sine, cosine, and tangent. Knowing SOH-CAH-TOA and the reciprocal relationships gives you all six trig ratios from any right triangle.