The image you uploaded contains right-angled triangles with missing side lengths that need to be calculated using trigonometric ratios (sine, cosine, and tangent). Let's solve each of the problems step by step following the instructions provided:

a) Triangle RQP

• Given:
  • Angle $\angle R = 32^\circ$
  • Hypotenuse $PQ = 53.2 \, \text{m}$
  • We need to find side $PR$ (adjacent) and side $RQ$ (opposite).

1. To find $PR$ (adjacent): Use cosine. $\cos(32^\circ) = \frac{PR}{PQ}$ Solving for $PR$: $PR = PQ \times \cos(32^\circ)$ $PR = 53.2 \times \cos(32^\circ) \approx 53.2 \times 0.848 \approx 45.11 \, \text{m}$

2. To find $RQ$ (opposite): Use sine. $\sin(32^\circ) = \frac{RQ}{PQ}$ Solving for $RQ$: $RQ = PQ \times \sin(32^\circ)$ $RQ = 53.2 \times \sin(32^\circ) \approx 53.2 \times 0.530 \approx 28.20 \, \text{m}$

b) Triangle DEF

• Given:
  • Angle $\angle E = 56^\circ$
  • Hypotenuse $DF = 73 \, \text{cm}$
  • We need to find side $d$ (adjacent) and side $EF$ (opposite).

1. To find $d$ (adjacent): Use cosine. $\cos(56^\circ) = \frac{d}{73}$ Solving for $d$: $d = 73 \times \cos(56^\circ)$ $d = 73 \times 0.559 \approx 40.81 \, \text{cm}$

2. To find $EF$ (opposite): Use sine. $\sin(56^\circ) = \frac{EF}{73}$ Solving for $EF$: $EF = 73 \times \sin(56^\circ)$ $EF = 73 \times 0.829 \approx 60.52 \, \text{cm}$

c) Triangle LMN

• Given:
  • Angle $\angle L = 37.5^\circ$
  • Adjacent $MN = 6 \, \text{m}$
  • We need to find side $n$ (opposite) and side $LN$ (hypotenuse).

1. To find $n$ (opposite): Use tangent. $\tan(37.5^\circ) = \frac{n}{6}$ Solving for $n$: $n = 6 \times \tan(37.5^\circ)$ $n = 6 \times 0.767 \approx 4.60 \, \text{m}$

2. To find $LN$ (hypotenuse): Use cosine. $\cos(37.5^\circ) = \frac{6}{LN}$ Solving for $LN$: $LN = \frac{6}{\cos(37.5^\circ)}$ $LN = \frac{6}{0.794} \approx 7.56 \, \text{m}$

Would you like more details or clarification on any step?

Here are 5 related questions to further your understanding:

1. What are the conditions for using sine, cosine, or tangent to find missing sides?
2. How do you calculate the angle of a right triangle if the opposite and adjacent sides are known?
3. Can you explain how the Pythagorean theorem is related to trigonometric ratios?
4. What would change in the calculations if the angle given is not between the adjacent and hypotenuse?
5. How do trigonometric ratios apply to non-right triangles?

Tip: Always ensure your calculator is in degree mode when working with angles measured in degrees!