sin 30° = cos x°, find x Easy

The question

Question: find x.

sin 30° = cos x°.

x is between 0 and 90.

1 Draw the triangle

sin and cos come from the sides of a right triangle.

So we draw one, with the angle 30° at the bottom.

The other small angle is at the top.

We do not know it yet.

2 Find the other angle

The three angles of a triangle add up to 180°.

The right angle takes 90°.

180 minus 90 is 90.

So the two small angles share 90°.

One small angle is 30°.

The other one gets the rest: 90 minus 30.

90 minus 30 is 60.

So the other angle is 60°.

3 Name the sides from each angle

Look at side a.

From the 30° angle, it is across: the opposite side.

We write opp, adj and hyp for short.

From the top angle, the same side a is next to it: the adjacent side.

Side c is the hypotenuse for both angles.

It is across from the right angle.

sin is the opposite side over the hypotenuse.

From 30°: a over c.

cos is the adjacent side over the hypotenuse.

From the top angle: a/c too.

Same two sides, same fraction.

So sin 30° = cos 60°.

This works in every right triangle.

The sin of one small angle is the cos of the other.

Together they make 90°.

4 Find x

The question says sin 30° = cos x°.

We found sin 30° = cos 60°.

So cos x° = cos 60°.

Careful!

sin and cos of the same angle are not equal.

sin 30° is 0.5, but cos 30° is about 0.866.

Between 0° and 90°, each angle has its own cos.

So x = 60.

5 Check

Let's check with the calculator.

Set the calculator to degrees first.

sin 30° is 0.5.

cos 60° is 0.5 too.

They are the same.

So x = 60 is right!

Let's look back.

First: draw the triangle.

Then: find the other angle.

Then: name the sides from each angle.

Then: find x.

Last: check.

▶ Watch it solved

A and B are the small angles of a right triangle, sin A = 4/5, find… Medium

The question

Question: find cos B.

A and B are the small angles of a right triangle, and sin A = 4/5.

1 Draw the triangle

A and B are the two small angles of one right triangle.

We draw it: A at the bottom, B at the top.

2 Name the sides from each angle

Look at side a.

From A, it is across: the opposite side.

We write opp, adj and hyp for short.

From B, the same side a is next to it: the adjacent side.

Side c is the hypotenuse for both angles.

sin is the opposite side over the hypotenuse.

From A: a over c.

cos is the adjacent side over the hypotenuse.

From B: a/c too.

Careful!

B is the other angle.

The side next to B is the side across from A.

Same two sides, same fraction.

So cos B = sin A.

3 Find cos B

sin A is 4/5.

So cos B = 4/5 too.

4 Check

Let's check with the calculator.

The sin⁻¹ button turns sin A back into the angle: A is about 53.1°.

So B is 90 minus 53.1: 36.9°.

cos 36.9° is about 0.8.

That is 4/5 again: 4 divided by 5 is 0.8.

So cos B = 4/5 is right!

Let's look back.

First: draw the triangle.

Then: name the sides from each angle.

Then: find cos B.

Last: check.

▶ Watch it solved

Solve cos(3x)° = sin(x + 14)° Harder

The question

Question: find x.

cos(3x)° = sin(x + 14)°.

Both angles are between 0 and 90.

1 Draw the triangle

sin and cos come from the sides of a right triangle.

So we draw one, with the angle (3x)° at the bottom.

The other small angle is at the top.

We call it y.

2 The two small angles

The three angles of a triangle add up to 180°.

The right angle takes 90°.

180 minus 90 is 90.

So the two small angles share 90°.

So 3x + y = 90.

3 Name the sides from each angle

Look at side b.

From the (3x)° angle, it is next to it: the adjacent side.

We write opp, adj and hyp for short.

From the top angle, the same side b is across: the opposite side.

Side c is the hypotenuse for both angles.

It is across from the right angle.

cos is the adjacent side over the hypotenuse.

From (3x)°: b over c.

sin is the opposite side over the hypotenuse.

From the top angle: b/c too.

Same two sides, same fraction.

So cos (3x)° = sin y°.

This works in every right triangle.

The cos of one small angle is the sin of the other.

Together they make 90°.

4 Write the equation

The question says cos(3x)° = sin(x + 14)°.

We found cos(3x)° = sin y°.

Between 0° and 90°, each angle has its own sin.

So y = x + 14.

In 3x + y = 90, put x + 14 in place of y: 3x + (x + 14) = 90.

Careful!

The two angles are not equal.

cos and sin are different.

The angles add up to 90.

A plus in front of a bracket: we can just drop the bracket.

5 Group like terms

Terms are the parts between the + and − signs.

Like terms are the same kind: same letters, same powers.

All the x terms go together.

3x + x = 4x.

Now we have 4x + 14 = 90.

We want x by itself.

Two things are stuck to x: times 4 and + 14.

First x was times 4.

Then + 14.

To take them off, we go backwards.

First we remove the + 14.

Then the times 4.

6 Remove the + 14

To remove + 14, we take away 14.

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

+ 14 and − 14 make zero.

Cross them out.

90 − 14 = 76.

Now we have 4x = 76.

7 Remove the 4

4x means 4 times x.

We want just x.

To remove times 4, we divide by 4.

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

4 divided by 4 is 1.

So just x is left.

76 divided by 4 is 19.

So x = 19!

8 Check

Let's check: put x = 19 in the angles.

3x is 57.

x + 14 is 33.

Now with the calculator.

Set the calculator to degrees first.

cos 57° is about 0.5446.

sin 33° is about 0.5446 too.

They are the same.

So x = 19 is right!

Let's look back.

First: draw the triangle.

Then: the two small angles.

Then: name the sides from each angle.

Then: write the equation.

Then: group like terms.

Then: remove the + 14.

Then: remove the 4.

Last: check.

▶ Watch it solved