Year 11 General Applications Of Trigonometry

Finding An Unknown Side In A Right-Angled Triangle

Q: What if the unknown side is on the bottom of the fraction?

Swap the unknown with the trig value. For example, sin 40 degrees equals 8 over x rearranges to x equals 8 divided by sin 40 degrees. On the calculator: divide the known side by the trig value.

Q: Why does my calculator give a tiny decimal answer?

Almost always: your calculator is in radian mode instead of degree mode. Switch to DEG (often via a MODE button or SETUP menu) and recalculate. Quick check: sin 30 degrees should give 0.5 — if it gives a negative number near 1 you're in radians.

Q: How many decimal places should I round to?

Whatever the question asks for. If a question says 'to 1 decimal place' round to 1 dp; if it says 'to the nearest cm' round to a whole number; if no instruction is given, 2 decimal places is usually safe.

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Theory

If one angle (not the right angle) and one side of a right-angled triangle are known, the other sides can be found using SOH-CAH-TOA. The unknown can sit on the top of the fraction (multiply across) or on the bottom (swap and divide). Always work in degree mode.

To find an unknown side in a right-angled triangle you need:

one acute angle (between $0^{\circ}$ and $90^{\circ}$ ) — not the right angle itself
one known side length

Label the three sides relative to the chosen angle $θ$ : opposite (across from $θ$ ), adjacent (next to $θ$ ) and hypotenuse (always opposite the right angle). Then pick the trig ratio that links the known side to the unknown one.

There are two situations depending on where the unknown sits:

Unknown on top (numerator) — multiply both sides by the denominator. Example: $\sin 40^{\circ} = \frac{x}{12}$ gives $x = 12 \sin 40^{\circ}$ .
Unknown on bottom (denominator) — swap the unknown with the trig value, then divide. Example: $\sin 40^{\circ} = \frac{8}{x}$ gives $x = \frac{8}{\sin 40^{\circ}}$ .

Unknown on top:

\tan 40^{\circ} = \frac{x}{12}

Unknown on bottom:

\sin 35^{\circ} = \frac{8}{h}

The three trig ratios

\sin θ = \frac{Opposite}{Hypotenuse} \cos θ = \frac{Adjacent}{Hypotenuse} \tan θ = \frac{Opposite}{Adjacent}

\sin θ = \frac{Opp}{Hyp}, \cos θ = \frac{Adj}{Hyp}, \tan θ = \frac{Opp}{Adj}

Rearranging to solve for the unknown

When the unknown side is on the top:

\sin θ = \frac{x}{c} \Rightarrow x = c \sin θ

x = c \sin θ

When the unknown side is on the bottom:

\sin θ = \frac{c}{x} \Rightarrow x = \frac{c}{\sin θ}

x = \frac{c}{\sin θ}

          Calculator setup: always check your calculator is in degree mode (DEG) before doing trig calculations. Working in radian or gradian mode gives wildly wrong answers.
        

How to find any unknown side

Label the three sides relative to the given angle: Opposite, Adjacent, Hypotenuse.
Identify the two sides the question involves — the known one and the unknown one.
Choose the trig ratio that uses those two sides: sin (Opp + Hyp), cos (Adj + Hyp), or tan (Opp + Adj).
Substitute the known values and solve for the unknown — multiply across if the unknown is on top, divide if it's on the bottom.

Example 1 — Unknown in numerator

A right triangle has an angle of

40^{\circ}

with the adjacent side

12

cm. Find the opposite side

x

(1 dp).

Solution

Sketch the triangle:

Opposite and adjacent are known/unknown, so use $\tan$ :

$\tan 40^{\circ}$	$=$	$\frac{x}{12}$
$x$	$=$	$12 \tan 40^{\circ}$
$x$	$\approx$	$10.1 cm$

x = 12 \tan 40 ° \approx 10.1 cm

Example 2 — Unknown in denominator

A right triangle has an angle of

35^{\circ}

with the opposite side

8

m. Find the hypotenuse

h

(1 dp).

Solution

Sketch the triangle:

Opposite and hypotenuse are involved, so use $\sin$ :

$\sin 35^{\circ}$	$=$	$\frac{8}{h}$
$h$	$=$	$\frac{8}{\sin 35^{\circ}}$
$h$	$\approx$	$13.9 m$

h = \frac{8}{\sin 35 °} \approx 13.9 m

Example 3 — Ladder problem

6

m ladder leans against a wall, making a

65^{\circ}

angle with the ground. Find the height it reaches up the wall (2 dp).

Solution

Sketch: the ladder is the hypotenuse, the height up the wall is opposite to $65^{\circ}$ :

$\sin 65^{\circ}$	$=$	$\frac{h}{6}$
$h$	$=$	$6 \sin 65^{\circ}$
$h$	$\approx$	$5.44 m$

h = 6 \sin 65 ° \approx 5.44 m

Example 4 — Flagpole shadow

A vertical flagpole casts a

12

m shadow when the sun's rays make a

35^{\circ}

angle with the ground. Find the height of the flagpole (1 dp).

Solution

Sketch: pole is opposite to $35^{\circ}$ , shadow is adjacent:

$\tan 35^{\circ}$	$=$	$\frac{h}{12}$
$h$	$=$	$12 \tan 35^{\circ}$
$h$	$\approx$	$8.4 m$

h = 12 \tan 35 ° \approx 8.4 m

Common pitfalls

Calculator in the wrong mode. Trig calculations must be in degree mode (DEG). Radians or gradians give nonsense answers.

Using the right angle as the working angle. The angle in your trig ratio must be one of the two acute angles (between

0^{\circ}

and

90^{\circ}

). The right angle is just a label.

Rounding too early. Carry full calculator precision through every intermediate step and only round at the very end.

Dropping the units. An answer of "

x \approx 11.7

" is incomplete — write

11.7

cm or

11.7

m. Marks are often lost here.

Frequently asked questions

How do I know whether to use sin, cos or tan?

Look at which two sides the question involves — the one you know and the one you're after. Pick sin for opposite and hypotenuse, cos for adjacent and hypotenuse, or tan for opposite and adjacent. The third side isn't part of the equation.

What if the unknown side is on the bottom of the fraction?

Swap the unknown with the trig value. For example, $\sin 40^{\circ} = \frac{8}{x}$ rearranges to $x = \frac{8}{\sin 40^{\circ}}$ . On the calculator: divide the known side by the trig value.

How do I find the hypotenuse using trig?

If you know an angle and either the opposite or adjacent side, the hypotenuse will end up on the bottom. Use $\sin θ = \frac{opp}{hyp}$ (so $hyp = \frac{opp}{\sin θ}$ ) or $\cos θ = \frac{adj}{hyp}$ (so $hyp = \frac{adj}{\cos θ}$ ).

Why does my calculator give a tiny decimal answer?

Almost always: your calculator is in radian mode instead of degree mode. Switch to DEG (often via a MODE button or SETUP menu) and recalculate. Quick check: $\sin 30^{\circ}$ should give $0.5$ — if it gives $- 0.988$ you're in radians.

Do I need to know all three sides to use trig?

No — that's the whole point. Trig lets you find an unknown side from just one angle and one side. You only need Pythagoras if you have two sides and no angle.

How many decimal places should I round to?

Whatever the question asks for. If a question says "to 1 decimal place" round to 1 dp; if it says "to the nearest cm" round to a whole number; if no instruction is given, 2 decimal places is usually safe.

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Finding An Angle In A Right-Angled Triangle

Finding An Unknown Side In A Right-Angled Triangle

📖 Theory

The three trig ratios

Rearranging to solve for the unknown

How to find any unknown side

Common pitfalls

Frequently asked questions

Theory