The hypotenuse is the longest side
sin A and cos A are never more than 1; sec A and cosec A are never less than 1.
Top over bottom
sin A = BC/AC. The top is a leg of the right triangle. The bottom is the hypotenuse.
The hypotenuse is the side across from the right angle, and it is the longest side.
Can sin A be bigger than 1?
What this lesson covers
The idea
Since the hypotenuse is the longest side of a right triangle, sin A and cos A never exceed 1, while sec A and cosec A are always greater than or equal to 1.
Top over bottom
sin A = BC/AC. The top is a leg of the right triangle. The bottom is the hypotenuse.
The hypotenuse is the side across from the right angle, and it is the longest side.
Can sin A be bigger than 1?
- No. A leg is never longer than the hypotenuse.
- Yes, in a very thin triangle.
- Yes, when the angle is large.
Swing the arm
The hypotenuse AC is an arm of fixed length 10. Swing it from flat to upright by dragging C along the dashed arc. The bars show sin A, cos A, cosec A and sec A. The dashed line marks 1.
A leg is never longer
The hypotenuse is the longest side, so sin A and cos A never exceed 1 (they are less than 1, or equal to 1 at the ends). sec A and cosec A are always greater than or equal to 1.
sin A = BC/AC and cos A = AB/AC put a leg over the hypotenuse, so the fraction is at most 1. sec A and cosec A are the flips of cos A and sin A, so they are at least 1.
Notes
The hypotenuse is the longest side of a right triangle. So sin A ≤ 1 and cos A ≤ 1, while sec A ≥ 1 and cosec A ≥ 1.
Check yourself
Which of these can not be the value of sin A?
Which of these can not be the value of sec A?
cosec A = 1.25. What is sin A?
Answer: 0.8
sin A = 1 ÷ 1.25 = 0.8. This is less than 1, as it must be.
A student writes sin A = 5/4 for a triangle she drew. What is wrong?
- 0.9. A leg can be 0.9 of the hypotenuse, so sin A can be 0.9.
- 1.2 — correct. Yes! sin A is a leg divided by the hypotenuse, so it cannot be more than 1.
- 0.5. A leg can be half the hypotenuse, so sin A can be 0.5.
- 0.8 — correct. Yes! sec A = hypotenuse / adjacent. The hypotenuse is the longest side, so sec A is at least 1.
- 1.5. sec A can be 1.5 when the hypotenuse is 1.5 times the adjacent side.
- 3. sec A can be 3 when the hypotenuse is 3 times the adjacent side.
- Nothing. sin A can be any number.. sin A is a leg divided by the hypotenuse, so it cannot be more than 1.
- The leg BC would be longer than the hypotenuse AC, which cannot happen. — correct. Yes! BC/AC = 5/4 would make BC longer than AC.
- sin A should be a whole number.. sin A is usually a fraction or a decimal, not a whole number.