Pick a Mode:

Throughout this webpage, we assume that x is strictly between 0 and π/2.

When entering an answer involving pi, you can just type "pi".

sin(x) < x


Angle:

Write a formula for the length A in terms of x:

Write a formula for the arclength B in terms of x:

Note that A represents a direct path from the common point to the baseline, while B represents an indirect path from the common point to the baseline.

xcos(x) < sin(x)


Angle: Crosshatch mode

Write a formula for the length A in terms of x:

Write a formula for the area of the triangle .

Write a formula for the area of the arc .

Write out the inequality that the area of the triangle is larger than the area of the arc. What do you need to multiply both sides by to get the inequality xcos(x) < sin(x)? Note that this quantity is positive for the given values of x.

xcos(x) < sin(x) < x

Note that for values of x close to 0, cos(x) is very close to 1, so the above inequality tells us that sin(x) is very close to x for small values of x.
y = x
y = sin(x)
y = xcos(x)

1-
x2
2
< cos(x)


Angle: Show bisector

Given that the length OC (the radius of the circle centered at O) is 1, write a formula for the length BC.

Select a correct argument that angle BAC is x/2.

Let y represent the length of the line segment AC. Note that y is less than

x, since x is the length of the arc AC. Select the correct trig function:

The length BC = y times

cos(x) < 1-
x2(cos(x))2
2

1 - cos(x) > (1-cos(x)) (1+cos(x))/2
=(sin(x))2/2
>
x2(cos(x))2
2

1-
x2
2
< cos(x) < 1-
x2(cos(x))2
2

Note that for values of x close to 0, cos(x) is very close to 1, so the above inequality tells us that cos(x) is very close to 1-
x2
2
for small values of x.
y = 1-
x2
2

y = sin(x)
y = 1-
x2(cos(x))2
2