KS3 · Maths

Right-angled triangle trigonometry: cosine

Your calculator's cos button isn't magic: it hands back the length of a side you could draw yourself.

Maths · Trigonometry

Drag C and watch cos θ
0.771.0040°90°ABC

adjacent AC = cos θ 0.77. hypotenuse AB 1.00. θ at A 40°. right angle at C 90°. Classification: right-angled. Relationship: hypotenuse = 1, so adjacent = cos θ

adjacent AC = cos θ0.77hypotenuse AB1.00θ at A40°right angle at C90°drag C round

right-angled

hypotenuse = 1, so adjacent = cos θ

AB is the hypotenuse and it is exactly 1 long. Drag C round and the right angle stays at C while θ (at A) changes. The adjacent side AC is the length we call cos θ. The readings come from the drawing, so treat them as estimates.

Maths · Trigonometry

Same angle, bigger triangle
0 cm1 cm2 cm3 cm4 cmdrag the hypotenuse

Hypotenuse h (cm): 4/4. Hypotenuse h 4 cm. Adjacent = h × 0.866 3.464 cm

Hypotenuse h4 cmAdjacent = h × 0.8663.464 cm

Take a triangle with hypotenuse 1 and an angle of 30°. A table of trig values gives cos 30° = 0.866, so its adjacent side is 0.866. A bigger right-angled triangle with the same angle is a similar triangle: every side is multiplied by the same scale factor, its hypotenuse. Drag the hypotenuse from 4 cm down to 1 cm and watch the adjacent side keep pace.

Maths · Algebra

One equation, three forms

Start from adjacent = h × cos θ. Divide both sides by the same thing and you get the other two forms. Pick a variant to see each one.

GoalGet h on its own
1
adjacent = h × cos θ

Our starting equation: the adjacent side is the hypotenuse times cos θ.

2
3

Step 1 of 3

Our starting equation: the adjacent side is the hypotenuse times cos θ.

Maths · Trigonometry

Which job is it?

Sort each situation by the thing you want to find.

Still to sort

Find the adjacent: multiply by cos θ (0)

You know the hypotenuse and θ.

Where the line is: Looks just like the hypotenuse job. Both know a length and an angle. Here the length you know is the hypotenuse.

Find the hypotenuse: divide by cos θ (0)

You know the adjacent and θ.

Where the line is: Here the length you know is the adjacent, and you are after the longest side.

Find the angle: use cos⁻¹ (0)

You know the adjacent and the hypotenuse.

Where the line is: No angle is given. You have two lengths and want θ.

6 of 6 still to sort.

Before any calculator, decide what you are trying to find. That choice picks the version of the formula.

Maths · Trigonometry

Your turn: choose the steps

Two triangles, two jobs. Triangle 1 has adjacent 22 cm and θ = 70°: find the hypotenuse. Triangle 2 has adjacent 38 cm and hypotenuse 89 cm: find θ. Give each answer to 1 decimal place.

  1. Triangle 1: we know the adjacent (22 cm) and the angle (70°), and we want the hypotenuse.
  2. missing step
Which line is step 2?

Maths · Trigonometry

Catch the slip

Find θ in a right-angled triangle with adjacent 6 cm and hypotenuse 10 cm.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The side you can watch shrink as the angle opens

What you need to know

  • The hypotenuse is the side opposite the right angle.
  • Next to both the right angle and θ sits the adjacent side; the opposite side faces θ.
  • Have a goA right-angled triangle has θ at the top corner and its right angle at the bottom left. Is the hypotenuse the left side, the bottom side or the sloping side?

    The sloping side.

    The hypotenuse is opposite the right angle, and the sloping side is the only one that does not touch the right-angle corner.

  • Draw a right-angled triangle with hypotenuse 1 unit, and the side adjacent to θ has length cos θ.
  • Any right-angled triangle is a scaled copy of that one, and the scale factor is its hypotenuse, h.
  • Scale cos θ by h and you get adjacent = h × cos θ.
  • Have a goIn a hypotenuse-1 triangle, cos θ = 0.7. A similar triangle has hypotenuse 10. How long is its adjacent side?

    7

    Every side is multiplied by the scale factor 10, so adjacent = h × cos θ = 10 × 0.7. Adding 10 instead would not keep the same shape.

  • Divide both sides by cos θ for h = adjacent ÷ cos θ, or by h for cos θ = adjacent ÷ hypotenuse.
  • Choose the version with your unknown as its subject: multiply by cos θ to find the adjacent, divide to find the hypotenuse.
  • To find an angle, use the inverse cosine: θ = cos⁻¹(adjacent ÷ hypotenuse), the cos⁻¹ button on a calculator.
  • Cosine is never greater than 1, and doubling an angle does not double its cosine.
  • Have a goSam announces, very confidently: "cos 30° = 0.866, so cos 60° is double that: 1.732!" Give one reason Sam must be wrong.

    Cosine is never greater than 1, so 1.732 can't be a cosine.

    Doubling the angle does not double its cosine either: cosine does not scale linearly with the angle.

  • Put your calculator in degrees mode, round only at the end, and check the hypotenuse comes out longest.

The big picture

Cosine links the hypotenuse, the side adjacent to an angle and the angle itself. In a triangle with hypotenuse 1, the adjacent side is cos θ. Scale that triangle by the hypotenuse h and you get adjacent = h × cos θ, which you can rearrange to find a hypotenuse or an angle.

Key points

1cos θ is the adjacent side of the right-angled triangle whose hypotenuse is 1.
2adjacent = h × cos θ, h = adjacent ÷ cos θ and cos θ = adjacent ÷ hypotenuse are one equation in three forms.
3To find an angle, θ = cos⁻¹(adjacent ÷ hypotenuse), with the calculator in degrees mode.
4Cosine is never greater than 1, and rounding happens only at the final stage.

Worked example

Problem

A 4.5 m ladder leans against a wall and makes an angle of 68° with the ground. How far is the foot of the ladder from the wall? Give your answer to 1 decimal place.

⚠ Watch out

Putting the fraction the wrong way up, hypotenuse ÷ adjacent. That gives a value above 1, and a calculator shows a maths error when you try cos⁻¹ of it. Cosine is adjacent ÷ hypotenuse, so it can never be greater than 1.

🧠

Memory hook

Cosine: the side next to θ goes on top, the hypotenuse goes underneath, and the answer never tops 1.

✓

Check yourself

Which side goes on top of the cosine fraction? You get cos θ = 1.4 — what went wrong? Hypotenuse and θ known: multiply or divide by cos θ?

Flashcards

(14)
Which side is the hypotenuse?
The side opposite the right angle.
Which side is the adjacent side?
The side next to both the right angle and the marked angle θ.
Which side is the opposite side?
The side opposite the marked angle θ.
In a right-angled triangle with hypotenuse 1, how long is the side adjacent to θ?
cos θ.
How is any right-angled triangle related to the hypotenuse-1 triangle with the same angle?
It is a scaled copy (a similar triangle). The scale factor is its hypotenuse, h.
Write the three forms of the cosine formula.
adjacent = h × cos θ; h = adjacent ÷ cos θ; cos θ = adjacent ÷ hypotenuse.
How are the two rearrangements derived?
By dividing both sides of h × cos θ = adjacent by h or by cos θ.
Which calculator button turns a cosine ratio back into an angle?
cos⁻¹ (arccos). Find the ratio adjacent ÷ hypotenuse first, then press cos⁻¹.
Which three pairs of facts let you use the cosine ratio?
Hypotenuse and angle; adjacent and angle; or hypotenuse and adjacent.
What is the greatest value cos θ can have?
1. Cosine is never greater than 1, so adjacent ÷ hypotenuse is the right way up.
If θ doubles, does cos θ double?
No. Cosine does not scale linearly with the angle.
A table of trig values or the cos button: what is the difference?
A table gives cosines for certain angles only (cos 30° = 0.866 to 3 d.p.). The cos button finds the cosine of any angle.
What must you check before using a calculator for trigonometry?
That it is in degrees mode, not radians or gradians. In degrees mode you don't type the ° symbol.
When do you round in a trigonometry calculation?
Only at the final stage. Then sense-check that the hypotenuse is longer than the adjacent.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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