KS3 · Maths
Choosing the right trigonometric ratio (SOHCAHTOA)
In a triangle with angles 58° and 32°, one side has two different names without moving an inch. Which one it answers to depends on where you stand.
Maths · Right-angled triangles
Which method? Sort the problem before you touch the calculator
Pick a problem, then the method that gets the answer most efficiently.
Still to sort
Sine (0)
opposite and hypotenuse: sin θ = opposite ÷ hypotenuse
Where the line is: Sine and cosine both use the hypotenuse. What splits them is the other side: across from the focus angle means sine.
Cosine (0)
adjacent and hypotenuse: cos θ = adjacent ÷ hypotenuse
Where the line is: Same hypotenuse partner as sine. The other side touches the focus angle (and is not the hypotenuse), so it is cosine.
Tangent (0)
opposite and adjacent: tan θ = opposite ÷ adjacent
Where the line is: The only one of the three that does not involve the hypotenuse at all.
Pythagoras' theorem (0)
two sides known, the third side wanted, no angle involved
Where the line is: Spot the difference from trigonometry: with one side and an angle you use a ratio; with two sides and the third wanted, Pythagoras may be more efficient.
Two questions decide it: which two sides are involved (one you know, one you want), and where they sit compared with the focus angle. Sort each problem under sine, cosine, tangent or Pythagoras.
A practical problem
Problem
You stand 3 m from a traffic light. Your eye height is 1.56 m and the angle of elevation of the top of the light is 52°. Estimate the height of the traffic light.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
What you need to know
- Name the sides from the focus angle, the angle you're using. How the triangle is turned doesn't matter.
- The horizontal edge isn't always the adjacent: the side opposite 58° is adjacent to the 32° in the same triangle.
Have a goSam says, "The bottom edge is always the adjacent. Always." In a triangle with angles 58° and 32°, the bottom edge is the side opposite the 58°. Taking the 58° as your focus angle, is Sam right?
No. From the 58°, that bottom edge is the opposite.
It's only adjacent when the 32° is the focus angle. The name comes from the focus angle, not from where the side sits on the page.
- Each ratio links two sides and an angle: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.
- Circle the side you know and the side you want. That pair picks the ratio.
Have a goYou know the side adjacent to the focus angle and you want the side opposite it. Circle that pair. Which ratio does it name?
Tangent: tan θ = opposite ÷ adjacent.
Neither of the two sides is the hypotenuse, so sine and cosine, which both use it, are out.
- Any of the three can work with enough steps, but one is normally the most efficient.
- Wanted side on top of the fraction? Multiply. Underneath? Divide.
Have a goYou have tan θ = 5 ÷ x and you want x. Multiply or divide, and what is x?
Divide: x = 5 ÷ tan θ.
x is underneath the fraction. Multiplying (5 × tan θ) is the tempting slip, and it only works when the wanted side is on top.
- Finding an angle? Form the ratio from the two known sides, then use the inverse function: arcsin, arccos or arctan.
- Two sides known and the third wanted? Pythagoras may be more efficient. One side and an angle? Use trigonometry.
- Keep unrounded values in later steps and round only at the end.
- Angle of elevation: measured up from the horizontal eye line. Angle of depression: measured down from it.
The big picture
To solve a right-angled triangle, name the sides from the focus angle, circle the side you know and the side you want, and let that pair choose sine, cosine or tangent. If two sides are known and you want the third, Pythagoras may be quicker. Then use the method to find a side or an angle, including in multi-step practical problems.
Key points
Worked example
Problem
A ramp up to a doorway 34 cm above the ground can have an incline of at most 4.8°. Using exactly 4.8°, how far from the wall does the ramp start?
⚠ Watch out
Calling the horizontal side the adjacent every time. Opposite and adjacent are named from the focus angle: switch angle and the two shorter sides swap names.
Memory hook
Circle what you know. Circle what you want. The focus angle names the sides, and the pair of sides names the ratio.
Check yourself
A right-angled triangle has a 20° angle. You know the hypotenuse and want the side adjacent to the 20°. Which method, and why not Pythagoras?
Flashcards
(15)What is the focus angle?
Which side is the hypotenuse?
Is the horizontal side always the adjacent?
sin θ = ?
cos θ = ?
tan θ = ?
How do you choose between sine, cosine and tangent?
You know the adjacent and the hypotenuse. Which ratio?
Wanted side on top of the ratio: multiply or divide? Underneath?
How do you find an angle when two sides are known?
When may Pythagoras be more efficient than trigonometry?
What do Pythagoras and the trig ratios both need?
In a multi-step problem, when do you round?
Angle of elevation or depression: which is measured down from the horizontal eye line?
How can you use trigonometry in a triangle with no right angle?
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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Keep me postedMore KS3 Maths topics
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