Let's solve each of these
SOHCAHTOA word problems step by step. We'll use trigonometric ratios:
-
SOH: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
-
CAH: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
-
TOA: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
---
Problem 1
> A wooden beam 24 feet long leans against a wall and makes an angle of $71^\circ$ with the ground. How high up the wall does the beam reach to the nearest foot?
We are given:
- Hypotenuse = 24 ft
- Angle with ground = $71^\circ$
- We want the
height (opposite side)
Use
SOH:
$$
\sin(71^\circ) = \frac{\text{opposite}}{24}
\Rightarrow \text{opposite} = 24 \cdot \sin(71^\circ)
$$
Calculate:
$$
\sin(71^\circ) \approx 0.9455
\Rightarrow 24 \cdot 0.9455 \approx 22.69 \text{ ft}
$$
Round to nearest foot:
23 feet
✔ Answer: B) 23 feet
---
Problem 2
> A ladder leaning against a building makes an angle of $65^\circ$ with the ground and reaches a point on the building 20 feet above the ground. What is the length of the ladder to the nearest foot?
Given:
- Opposite = 20 ft
- Angle = $65^\circ$
- Find hypotenuse (ladder length)
Use
SOH:
$$
\sin(65^\circ) = \frac{20}{\text{hypotenuse}}
\Rightarrow \text{hypotenuse} = \frac{20}{\sin(65^\circ)}
$$
$$
\sin(65^\circ) \approx 0.9063
\Rightarrow \frac{20}{0.9063} \approx 22.07 \text{ ft}
$$
Round to nearest foot:
22 feet
✔ Answer: D) 22 feet
---
Problem 3
> An airplane climbs at an angle of $13^\circ$ with the ground. What is the distance it has traveled (to the nearest hundred feet) when it has attained an altitude of 400 feet?
Given:
- Opposite = 400 ft (altitude)
- Angle = $13^\circ$
- Find hypotenuse (distance traveled)
Use
SOH:
$$
\sin(13^\circ) = \frac{400}{\text{hypotenuse}}
\Rightarrow \text{hypotenuse} = \frac{400}{\sin(13^\circ)}
$$
$$
\sin(13^\circ) \approx 0.2225
\Rightarrow \frac{400}{0.2225} \approx 1,800 \text{ ft}
$$
To nearest hundred feet:
1,800 feet
✔ Answer: C) 1,800 feet
---
Problem 4
> A 20-foot pole leaning against a wall reaches a point 18 feet above the ground. What is the angle which the pole makes with the ground to the nearest degree?
Given:
- Hypotenuse = 20 ft
- Opposite = 18 ft
- Find angle $\theta$
Use
SOH:
$$
\sin(\theta) = \frac{18}{20} = 0.9
\Rightarrow \theta = \sin^{-1}(0.9)
$$
$$
\sin^{-1}(0.9) \approx 64.16^\circ
$$
Round to nearest degree:
64°
✔ Answer: A) 64°
---
Problem 5
> When the plane had flown 4,150 feet from the airport where it had taken off, it had covered a horizontal distance of 3,660 feet. What is the angle at which the plane rose from the ground to the nearest degree?
We have:
- Hypotenuse = 4,150 ft (total distance flown)
- Adjacent = 3,660 ft (horizontal distance)
- Want angle $\theta$ between ground and flight path
Use
CAH:
$$
\cos(\theta) = \frac{3,660}{4,150} \approx 0.8819
\Rightarrow \theta = \cos^{-1}(0.8819)
$$
$$
\cos^{-1}(0.8819) \approx 28.1^\circ
$$
Round to nearest degree:
28°
✔ Answer: A) 28°
---
Problem 6
> At a point on the ground 46 feet from the foot of a tree, the angle of elevation of the top of the tree is $48^\circ$. What is the height of the tree to the nearest foot?
Given:
- Adjacent = 46 ft
- Angle = $48^\circ$
- Find opposite (height of tree)
Use
TOA:
$$
\tan(48^\circ) = \frac{\text{opposite}}{46}
\Rightarrow \text{opposite} = 46 \cdot \tan(48^\circ)
$$
$$
\tan(48^\circ) \approx 1.1106
\Rightarrow 46 \cdot 1.1106 \approx 50.98 \text{ ft}
$$
Round to nearest foot:
51 feet
✔ Answer: C) 51 feet
---
✔ Final Answers:
1)
B) 23 feet
2)
D) 22 feet
3)
C) 1,800 feet
4)
A) 64°
5)
A) 28°
6)
C) 51 feet
Let me know if you'd like diagrams or further explanation!
Parent Tip: Review the logic above to help your child master the concept of sohcahtoa word problems worksheet.
