Exact value of sin 30° Easy

The question

Question: find the exact value of sin 30°.

1 Draw the unit circle

The unit circle has its center at 0 and radius 1.

We turn 30° from the positive x-axis, against the clock.

The ray meets the circle at the point P.

Here is the key fact.

P is the point (cos 30°, sin 30°).

Across is cos.

Up is sin.

2 Use the special triangle

Draw a line from P straight down to the x-axis.

It makes a right triangle.

Its longest side goes from the center to P.

It is a radius: 1.

The angles are 30°, 60° and 90°.

The short side x is across from the 30° angle.

From the special triangles lesson: the sides are x, x root 3 and 2x.

The longest side, 2x, is 1.

So 2x = 1.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2 divided by 2 is 1.

So just x is left.

So x = 1/2: the short side.

The other side is x root 3: 1/2 times √3.

That is √3/2.

3 Read the point

Across, from the center to under P: √3/2.

Up, from the x-axis to P: 1/2.

So P = (√3/2, 1/2).

Careful!

Sin is the UP side, not the across side.

Up from the x-axis to P is 1/2.

Sin is the up number.

So sin 30° = 1/2.

4 Check

Let's check: P is on the circle, so across squared plus up squared must be 1.

Pythagoras, with the longest side 1.

It is 1.

So the point is right, and sin 30° = 1/2 is right!

Let's look back.

First: draw the unit circle.

Then: use the special triangle.

Then: read the point.

Last: check.

▶ Watch it solved

Exact value of tan 30° Medium

The question

Question: find the exact value of tan 30°.

1 Draw the unit circle

The unit circle has its center at 0 and radius 1.

We turn 30° from the positive x-axis, against the clock.

The ray meets the circle at the point P.

Here is the key fact.

P is the point (cos 30°, sin 30°).

Across is cos.

Up is sin.

2 Use the special triangle

Draw a line from P straight down to the x-axis.

It makes a right triangle.

Its longest side goes from the center to P.

It is a radius: 1.

The angles are 30°, 60° and 90°.

The short side x is across from the 30° angle.

From the special triangles lesson: the sides are x, x root 3 and 2x.

The longest side, 2x, is 1.

So 2x = 1.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2 divided by 2 is 1.

So just x is left.

So x = 1/2: the short side.

The other side is x root 3: 1/2 times √3.

That is √3/2.

3 Read the point

Across, from the center to under P: √3/2.

Up, from the x-axis to P: 1/2.

So P = (√3/2, 1/2).

Careful!

Tan is sin over cos: the up number divided by the across number.

4 Divide

Tan is sin over cos: up divided by across.

Now put the numbers in: sin = 1/2 and cos = √3/2.

To divide by a fraction, we multiply by its flip.

A 2 at the bottom and a 2 on top: 2/2 is 1.

Cross them out.

What is left: 1 over √3.

Books write it with no root at the bottom: times √3 on top and bottom.

√3 times √3 is 3.

So tan 30° = √3/3.

5 Check

Let's check: tan times cos must give sin back.

√3/3 times √3/2.

We get 1/2: that is sin 30°.

So it is right!

Let's look back.

First: draw the unit circle.

Then: use the special triangle.

Then: read the point.

Then: divide.

Last: check.

▶ Watch it solved

Exact value of sin 90° Harder

The question

Question: find the exact value of sin 90°.

1 Draw the unit circle

The unit circle has its center at 0 and radius 1.

We turn 90° from the positive x-axis, against the clock.

The ray meets the circle at the point P.

Here is the key fact.

P is the point (cos 90°, sin 90°).

Across is cos.

Up is sin.

2 Read the point

P is on the y-axis, one step from the center: up.

So P = (0, 1).

Careful!

The first number of the point is cos.

The second is sin.

Not the other way round.

Sin is the up number.

So sin 90° = 1.

3 Check

Let's check with the calculator: sin 90°.

The calculator says 1 too.

So it is right!

Let's look back.

First: draw the unit circle.

Then: read the point.

Last: check.

▶ Watch it solved