Which one is Greater: Sin 30° or Sin 40°?

Question

My homework says do it without a calculator or a table.

How the hell are you supposed to know which one is greater?

Answer 1

the sine curve goes like this (roughly...remember it is a smooth curve....):

.....__ value 1
... /....\
../.......\
/----.-----\------------ zero line
.............\
...............\
...............---- value -1


sin0=0, sin90=1, sin180=0, sin 270= -1, sin360=0 (back to 0)

so sine increases from 0 to 90 (value 0 to 1)
it decreases from 90+ to 180 (value 1 to 0)
it decreases from 180+ to 270 (value 0 to -1)
it increases from 270+ to 360 (value -1 to 0)

then the cycle repeats....

so sin40 is bigger than sin30 (increasing region)

sin140 is smaller than sin130 (decreasing region)

Answer 2

It's asking you to work it out based on what you should already know or should have observed about the value of sine as the angle it describes approaches 90 degrees (you are working in degrees, yes?). Anyhow, the sine increases as the angle opens up from zero degrees and gets closer to 90 degrees... sin 0 = 0 and sin 90 = 1 ... It'll help you if you can remember that. Therefore...


since 40 is closer to 90 than 30, its sine must be larger.

Answer 3

Remember that the sine of an angle in a right triangle is opposite side over hypotenuse. So as the angle gets larger, so does the opposite side. Therefore, opposite over hypotenuse increases. So sine 40 is larger than sine 30.

Another way to look at it is to know that sine 0 = 0 and sine 90 = 1. Therefore, the value of the sine goes from 0 to 1 as the angle goes from 0 to 90. So the sine value increases as the angle increases.

Note: This is not true for angles from 90 to 270.

This discussion refers to angles in degrees.

Answer 4

sin 40 is greater because:
sin 30 = .5
sin 40 = .64278