Worked examples
The question
Question: find tan A, when sin A = 3/5.
sin A = 35 tan A = ?
1 Draw the triangle
sin, cos and tan belong to an angle of a right triangle.
So we draw a right triangle with the angle A.
2 Use what sin A means
The opposite side is across from A.
We write opp, adj and hyp for short.
The adjacent side is next to A.
It is not the hypotenuse.
The hypotenuse is the longest side.
It is across from the right angle.
sin A is the opposite side over the hypotenuse.
sin A = opphyp
The question says sin A = 3/5.
So opp over hyp is 3/5.
opphyp = 35
Every right triangle with the angle A has the same shape.
A bigger one could have 6 and 10.
6/10 is 3/5 too.
610 = 35
So we can use the small one: opp = 3 and hyp = 5.
3 Find the missing side
The adjacent side is missing.
We call it x.
Pythagoras: in a right triangle, the two short sides squared add up to the hypotenuse squared.
opp2 + adj2 = hyp2
Now put the numbers in: opp = 3, adj = x and hyp = 5.
32 + x2 = 52
3² means 3 times 3.
That is 9.
32 = 3 × 3 = 9
5² means 5 times 5.
That is 25.
52 = 5 × 5 = 25
Write the squares: 9 + x² = 25.
9 + x2 = 25
We want x² by itself.
The 9 is in the way.
To remove 9, we take away 9.
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.
9 + x2 − 9 = 25 − 9
9 and − 9 make zero.
Cross them out.
25 minus 9 is 16.
x2 = 16
x² is 16.
So x times x is 16.
That means x is the square root of 16.
x = √16
The square root of 16 is the plus number that times itself makes 16.
4 times 4 is 16.
−4 times −4 is 16 too, but the √ sign always means the plus one.
4 × 4 = 16
So x = 4.
x = 4
4 Write tan A
tan A is the opposite side over the adjacent side.
tan A = oppadj
Put in opp = 3 and adj = 4.
So tan A = 3/4.
tan A = 34
Careful!
The hypotenuse is never in tan.
tan uses the two short sides: opp and adj.
Careful: not tan A = 35 ✗
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
The sin⁻¹ button turns sin A back into the angle: A is about 36.9°.
A = sin−1(3/5) ≈ 36.9°
tan 36.9° is about 0.75.
tan 36.9° ≈ 0.75
And 3/4 is 0.75.
34 = 0.75
The same.
So tan A = 3/4 is right!
Let's look back.
First: draw the triangle.
Then: use what sin A means.
Then: find the missing side.
Then: write tan A.
Last: check.
▶ Watch it solved
The question
Question: find tan A, when sin A = 7/25.
sin A = 725 tan A = ?
1 Draw the triangle
sin, cos and tan belong to an angle of a right triangle.
So we draw a right triangle with the angle A.
2 Use what sin A means
The opposite side is across from A.
We write opp, adj and hyp for short.
The adjacent side is next to A.
It is not the hypotenuse.
The hypotenuse is the longest side.
It is across from the right angle.
sin A is the opposite side over the hypotenuse.
sin A = opphyp
The question says sin A = 7/25.
So opp over hyp is 7/25.
opphyp = 725
Every right triangle with the angle A has the same shape.
A bigger one could have 14 and 50.
14/50 is 7/25 too.
1450 = 725
So we can use the small one: opp = 7 and hyp = 25.
3 Find the missing side
The adjacent side is missing.
We call it x.
Pythagoras: in a right triangle, the two short sides squared add up to the hypotenuse squared.
opp2 + adj2 = hyp2
Now put the numbers in: opp = 7, adj = x and hyp = 25.
72 + x2 = 252
7² means 7 times 7.
That is 49.
72 = 7 × 7 = 49
25² means 25 times 25.
That is 625.
252 = 25 × 25 = 625
Write the squares: 49 + x² = 625.
49 + x2 = 625
We want x² by itself.
The 49 is in the way.
To remove 49, we take away 49.
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.
49 + x2 − 49 = 625 − 49
49 and − 49 make zero.
Cross them out.
625 minus 49 is 576.
x2 = 576
x² is 576.
So x times x is 576.
That means x is the square root of 576.
x = √576
The square root of 576 is the plus number that times itself makes 576.
24 times 24 is 576.
−24 times −24 is 576 too, but the √ sign always means the plus one.
24 × 24 = 576
So x = 24.
x = 24
4 Write tan A
tan A is the opposite side over the adjacent side.
tan A = oppadj
Put in opp = 7 and adj = 24.
So tan A = 7/24.
tan A = 724
Careful!
The hypotenuse is never in tan.
tan uses the two short sides: opp and adj.
Careful: not tan A = 725 ✗
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
The sin⁻¹ button turns sin A back into the angle: A is about 16.3°.
A = sin−1(7/25) ≈ 16.3°
tan 16.3° is about 0.29.
tan 16.3° ≈ 0.29
And 7/24 is about 0.29.
724 ≈ 0.29
The same.
So tan A = 7/24 is right!
Let's look back.
First: draw the triangle.
Then: use what sin A means.
Then: find the missing side.
Then: write tan A.
Last: check.
▶ Watch it solved
The question
Question: find sin A, when cos A = 2/3.
cos A = 23 sin A = ?
1 Draw the triangle
sin, cos and tan belong to an angle of a right triangle.
So we draw a right triangle with the angle A.
2 Use what cos A means
The opposite side is across from A.
We write opp, adj and hyp for short.
The adjacent side is next to A.
It is not the hypotenuse.
The hypotenuse is the longest side.
It is across from the right angle.
cos A is the adjacent side over the hypotenuse.
cos A = adjhyp
The question says cos A = 2/3.
So adj over hyp is 2/3.
adjhyp = 23
Every right triangle with the angle A has the same shape.
A bigger one could have 4 and 6.
4/6 is 2/3 too.
46 = 23
So we can use the small one: adj = 2 and hyp = 3.
3 Find the missing side
The opposite side is missing.
We call it x.
Pythagoras: in a right triangle, the two short sides squared add up to the hypotenuse squared.
opp2 + adj2 = hyp2
Now put the numbers in: opp = x, adj = 2 and hyp = 3.
x2 + 22 = 32
2² means 2 times 2.
That is 4.
22 = 2 × 2 = 4
3² means 3 times 3.
That is 9.
32 = 3 × 3 = 9
Write the squares: x² + 4 = 9.
x2 + 4 = 9
We want x² by itself.
The 4 is in the way.
To remove 4, we take away 4.
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.
x2 + 4 − 4 = 9 − 4
4 and − 4 make zero.
Cross them out.
9 minus 4 is 5.
x2 = 5
x² is 5.
So x times x is 5.
That means x is the square root of 5.
x = √5
5 is not a square number.
Its root is a decimal that never ends.
So we keep √5.
That is the exact length.
Careful!
We cannot just take away the sides.
Pythagoras works with the squares: 9 − 4 = 5.
Then the square root.
Careful: not x = 3 − 2 = 1 ✗
4 Write sin A
sin A is the opposite side over the hypotenuse.
sin A = opphyp
Put in opp = √5 and hyp = 3.
So sin A = √5/3.
sin A = √53
5 Check
Let's check with the calculator.
Set the calculator to degrees first.
The cos⁻¹ button turns cos A back into the angle: A is about 48.2°.
A = cos−1(2/3) ≈ 48.2°
sin 48.2° is about 0.75.
sin 48.2° ≈ 0.75
And √5/3 is about 0.75.
√53 ≈ 0.75
The same.
So sin A = √5/3 is right!
Let's look back.
First: draw the triangle.
Then: use what cos A means.
Then: find the missing side.
Then: write sin A.
Last: check.
▶ Watch it solved
