Find an unknown side of a right-angled triangle lesson | Year 10 Mathematics (Pythagoras and trigonometry, Unknown side in the denominator)

Find a denominator side

A denominator length is isolated by clearing the denominator and then dividing by the ratio value

Choose the pair Identify the known and wanted sides from the marked angle. Write the matching ratio before deciding which operation to use \(\displaystyle \begin{aligned}\sin\theta&=\frac{o}{h}\\\cos\theta&=\frac{a}{h}\\\tan\theta&=\frac{o}{a}\end{aligned}\)
Clear the unknown denominator The unknown length is positive. Multiply both sides by it, then divide both sides by the nonzero ratio value \(\displaystyle \begin{aligned}\sin\theta&=\frac{o}{x}\\x\sin\theta&=o\\x&=\frac{o}{\sin\theta}\end{aligned}\)
Preserve precision and check Use degree mode and round only the final length. An unknown hypotenuse must exceed the given leg. Dividing by tangent can increase or decrease a length, depending on the angle

An opposite leg is 12 cm and the acute angle is 38°. Find the hypotenuse x to two decimal places

\sin38^{\circ}=\frac{12}{x}

\(\displaystyle x\sin38^{\circ}=12\)Multiply both sides by the positive unknown length
\(\displaystyle x=\frac{12}{\sin38^{\circ}}\)Divide both sides by the nonzero sine value
\(\displaystyle x\approx19.491230\text{ cm}\)Evaluate in degree mode without intermediate rounding
\(\displaystyle x\approx19.49\text{ cm}\)Round once and check that the hypotenuse exceeds 12 cm

Multiplying by the ratio too soon

✕\(\displaystyle x=12\sin38^{\circ}\)✓\(\displaystyle x=\frac{12}{\sin38^{\circ}}\)

After clearing the unknown denominator, divide both sides by sine. Multiplication would make this hypotenuse shorter than the known leg

Using a rounded ratio

✕\(\displaystyle x=12\div0.62\)✓\(\displaystyle x=12\div\sin38^{\circ}\)

The ratio is dimensionless. Preserve its full calculator value until the final length

standardFind a denominator length with sine, cosine or tangent to two decimal places

examSelect the correct denominator length among distinct operation and ratio errors

Why does division appear?+

Multiplying by the unknown clears the denominator, but leaves the ratio as a factor of the unknown. Divide both sides by that factor

Does division always make the side longer?+

No. Acute sine and cosine are less than one, so dividing by them gives a larger hypotenuse. Tangent can exceed one, giving a shorter adjacent leg

Should I round the ratio first?+

No. Keep the full degree-mode value and round only the final required length

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 9: Pythagoras and trigonometry