Angle of elevation 35, 20 m from the tree, find the height Easy

The question

Question: find the height x of the tree.

The angle of elevation is 35°.

We stand 20 m from the tree.

The angle of elevation is at the eye.

It goes up from the flat ground to the line of sight.

1 Name the sides

We look from the angle 35°.

Every side gets a name from there.

The hypotenuse H is the longest side.

It is across from the right angle.

The opposite side O is across from our angle.

The adjacent side A is next to our angle.

It is not the hypotenuse.

2 Choose the ratio

SOH CAH TOA helps us remember.

Sin is O over H.

Cos is A over H.

Tan is O over A.

We have the adjacent and we want the opposite: A and O.

TOA has them.

So we use tan.

Careful!

The 20 m is along the ground: the adjacent side, not the hypotenuse.

So it is tan, not sin.

3 Write the equation

Now put the numbers in: A = 20 m and O = x.

4 Find x

x is on top, over 20.

To remove the 20 at the bottom, we multiply by 20.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

20/20 is 1.

So just x is left.

So x = 20 × tan 35°.

The calculator gives tan 35°: about 0.7002.

Put the number in.

20 times 0.7002 is about 14.004.

To the nearest tenth, x ≈ 14.0 m.

5 Check

Let's check: put x back in O over A.

We get 0.7.

tan 35° is about 0.7002.

Very close: the small gap is only from rounding.

So x is right!

Let's look back.

First: name the sides.

Then: choose the ratio.

Then: write the equation.

Then: find x.

Last: check.

▶ Watch it solved

Angle of elevation 40, kite string 30 m, find the height Medium

The question

Question: find the height x of the kite.

The angle of elevation is 40°.

The kite string is 30 m long.

The angle of elevation is at the eye.

It goes up from the flat ground to the line of sight.

1 Name the sides

We look from the angle 40°.

Every side gets a name from there.

The hypotenuse H is the longest side.

It is across from the right angle.

The opposite side O is across from our angle.

The adjacent side A is next to our angle.

It is not the hypotenuse.

2 Choose the ratio

SOH CAH TOA helps us remember.

Sin is O over H.

Cos is A over H.

Tan is O over A.

We have the hypotenuse and we want the opposite: H and O.

SOH has them.

So we use sin.

Careful!

The 30 m is the slanted side: the hypotenuse, not the ground.

So it is sin, not tan.

3 Write the equation

Now put the numbers in: H = 30 m and O = x.

4 Find x

x is on top, over 30.

To remove the 30 at the bottom, we multiply by 30.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

30/30 is 1.

So just x is left.

So x = 30 × sin 40°.

The calculator gives sin 40°: about 0.6428.

Put the number in.

30 times 0.6428 is about 19.284.

To the nearest tenth, x ≈ 19.3 m.

5 Check

Let's check: put x back in O over H.

We get 0.6433.

sin 40° is about 0.6428.

Very close: the small gap is only from rounding.

So x is right!

Let's look back.

First: name the sides.

Then: choose the ratio.

Then: write the equation.

Then: find x.

Last: check.

▶ Watch it solved

Angle of depression 40, tower 60 m tall, find the distance to the car Harder

The question

Question: find the distance x to the car.

The angle of depression is 40°.

The tower is 60 m tall.

1 Move the angle down

The angle of depression is at the top.

It goes down from the flat dashed line to the line of sight.

The flat line and the ground are parallel.

The line of sight crosses both.

So the angle at the car is the same: alternate angles, a Z shape.

So the angle at the car is 40° too.

Alternate angles are equal.

2 Name the sides

We look from the angle 40°.

Every side gets a name from there.

The hypotenuse H is the longest side.

It is across from the right angle.

The opposite side O is across from our angle.

The adjacent side A is next to our angle.

It is not the hypotenuse.

3 Choose the ratio

SOH CAH TOA helps us remember.

Sin is O over H.

Cos is A over H.

Tan is O over A.

We have the opposite and we want the adjacent: O and A.

TOA has them.

So we use tan.

Careful!

The angle of depression is between the flat line and the line of sight.

Inside the triangle at the top, the angle is 50°.

4 Write the equation

Now put the numbers in: O = 60 m and A = x.

5 Find x

x is at the bottom.

First we multiply by x.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

x/x is 1.

Only the top is left.

Now we have x × tan 40° = 60.

To remove times tan 40°, we divide by it.

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

It divides by itself: 1.

So just x is left.

The calculator gives tan 40°: about 0.8391.

Put the number in.

60 divided by 0.8391 is about 71.505, with the calculator.

To the nearest tenth, x ≈ 71.5 m.

6 Check

Let's check: put x back in O over A.

We get 0.8392.

tan 40° is about 0.8391.

Very close: the small gap is only from rounding.

So x is right!

Let's look back.

First: move the angle down.

Then: name the sides.

Then: choose the ratio.

Then: write the equation.

Then: find x.

Last: check.

▶ Watch it solved