Sohcahtoa word problems worksheet with answers pdf: Fill out ... - Free Printable

Here are the step-by-step solutions for each problem on your worksheet. These problems use trigonometry, specifically SOH CAH TOA (Sine, Cosine, Tangent), to find missing sides or angles in right triangles.

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?

* Identify: You have the hypotenuse (the beam, 24 ft) and the angle ($71^\circ$). You need to find the side opposite the angle (height on the wall).
* Choose Ratio: Sine = Opposite / Hypotenuse.
* Equation: $\sin(71^\circ) = x / 24$
* Solve: Multiply both sides by 24.
$$x = 24 \times \sin(71^\circ)$$
$$x \approx 24 \times 0.9455$$
$$x \approx 22.69$$
* Round: To the nearest foot, this is 23 feet.

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?

* Identify: You have the side opposite the angle (height, 20 ft) and the angle ($65^\circ$). You need to find the hypotenuse (ladder length).
* Choose Ratio: Sine = Opposite / Hypotenuse.
* Equation: $\sin(65^\circ) = 20 / x$
* Solve: Rearrange to solve for $x$.
$$x = 20 / \sin(65^\circ)$$
$$x \approx 20 / 0.9063$$
$$x \approx 22.06$$
* Round: To the nearest foot, this is 22 feet.

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?

* Identify: You have the side opposite the angle (altitude, 400 ft) and the angle ($13^\circ$). You need to find the hypotenuse (distance traveled).
* Choose Ratio: Sine = Opposite / Hypotenuse.
* Equation: $\sin(13^\circ) = 400 / x$
* Solve: Rearrange to solve for $x$.
$$x = 400 / \sin(13^\circ)$$
$$x \approx 400 / 0.22495$$
$$x \approx 1778.17$$
* Round: To the nearest hundred feet, 1778 rounds up to 1,800 feet.

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?

* Identify: You have the hypotenuse (pole, 20 ft) and the side opposite the angle (height, 18 ft). You need to find the angle.
* Choose Ratio: Sine = Opposite / Hypotenuse.
* Equation: $\sin(\theta) = 18 / 20$
* Solve: Use the inverse sine function ($\sin^{-1}$).
$$\theta = \sin^{-1}(0.9)$$
$$\theta \approx 64.15^\circ$$
* Round: To the nearest degree, this is $64^\circ$.

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?

* Identify: You have the hypotenuse (flight path, 4,150 ft) and the adjacent side (horizontal distance, 3,660 ft). You need to find the angle.
* Choose Ratio: Cosine = Adjacent / Hypotenuse.
* Equation: $\cos(\theta) = 3660 / 4150$
* Solve: Use the inverse cosine function ($\cos^{-1}$).
$$\theta = \cos^{-1}(0.8819...)$$
$$\theta \approx 28.12^\circ$$
* Round: To the nearest degree, this is $28^\circ$.

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?

* Identify: You have the adjacent side (distance from tree, 46 ft) and the angle ($48^\circ$). You need to find the opposite side (height of tree).
* Choose Ratio: Tangent = Opposite / Adjacent.
* Equation: $\tan(48^\circ) = x / 46$
* Solve: Multiply both sides by 46.
$$x = 46 \times \tan(48^\circ)$$
$$x \approx 46 \times 1.1106$$
$$x \approx 51.08$$
* Round: To the nearest foot, this is 51 feet.

Final Answer:
1) B
2) D
3) C
4) A
5) A
6) C