# Trigonometry lighthouse boat problem

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• Nate1
In summary: \cos\left(\theta_1\right)=\frac{x+20}{d_1}\implies d_1=(x+20)\sec\left(\theta_1\right)\cos\left(\theta_2\right)=\frac{x}{d_2}\implies d_2=x\sec\left(\theta_2\right)d_1=x\cos\left(\theta_1\right)+3.1677d_2=x\cos\left(\theta_2\right)+3.1677d_1=58.47+3.1677d_2=
Nate1
A boat is some distance away from a small lighthouse. The angle of elevation from the boat is 9 degrees. If the boat moves forward 20m, the angle is now 12. Assume the lighthouse is at a rightangle to the boat.

Calculate the distance from the boat to the top of the lighthouse before and after it moves
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Hi Nate and welcome to MHB!

Any thoughts on how to begin?

a/sinA = b/sinB = c/sinC

180 - 12 = 168

180 - (12+90) = 78
180 - (9+90) = 81
81 - 78 = 3 (top corner)

a/sin168 = 20/sin3
a = (20 x sin168)/sin3
a = 79.45

c/sin9 = 79.45/sin168
c = (79.45 x sin9)/sin168
c = 59.78

Correct?

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I would let:

$$\displaystyle h$$ = the height of the lighthouse in meters

$$\displaystyle \theta_1$$ = the initial angle of inclination

$$\displaystyle \theta_2$$ = the final angle of inclination

$$\displaystyle x$$ = the final distance to the lighthouse in meters

Now, we may state:

$$\displaystyle \tan\left(\theta_1\right)=\frac{h}{x+20}$$

$$\displaystyle \tan\left(\theta_2\right)=\frac{h}{x}$$

Now, these equations imply:

$$\displaystyle h=x\tan\left(\theta_2\right)=(x+20)\tan\left(\theta_1\right)$$

Or:

$$\displaystyle x\tan\left(\theta_2\right)=(x+20)\tan\left(\theta_1\right)$$

Now, solve this for $$x$$...what do you get?

x0.2126=x0.1584+3.1677
x0.0542=3.1677
x=58.47

Nate said:
x0.2126=x0.1584+3.1677
x0.0542=3.1677
x=58.47

I recommend solving the equation completely, then then plugging in the given values and rounding only at the very end.

$$\displaystyle x\tan\left(\theta_2\right)=(x+20)\tan\left(\theta_1\right)$$

$$\displaystyle x=\frac{20\tan\left(\theta_1\right)}{\tan\left(\theta_2\right)-\tan\left(\theta_1\right)}$$

Now plug in the given angles of inclination:

$$\displaystyle x=\frac{20\tan\left(9^{\circ}\right)}{\tan\left(12^{\circ}\right)-\tan\left(9^{\circ}\right)}\approx58.47452024794465$$

So, initially the boat was about 78.5 m from the lighthouse, and finally it was about 58.5 m from the lighthouse.

So which methed is more accurate?

Nate said:
So which methed is more accurate?

When working problems involving numeric data, it is best to only round at the end, and not use rounded values in intermediary steps, as this can sometimes magnify rounding errors. Here it didn't make much difference, if any, but it is good practice to only round once, at the end. :)

Sorry, by which method I meant the one you are using or the sin rule (seen above)

MarkFL said:
So, initially the boat was about 78.5 m from the lighthouse, and finally it was about 58.5 m from the lighthouse.

The problem is asking for the distances from the boat to the top of the lighthouse.

Olinguito said:
The problem is asking for the distances from the boat to the top of the lighthouse.

If we let $$d_1$$ be the initial distance from the eye of the observer to the top of the lighthouse, and $$d_2$$ be the final distance, and armed with the value of $$x$$, we could state:

$$\displaystyle \cos\left(\theta_1\right)=\frac{x+20}{d_1}\implies d_1=(x+20)\sec\left(\theta_1\right)$$

$$\displaystyle \cos\left(\theta_2\right)=\frac{x}{d_2}\implies d_2=x\sec\left(\theta_2\right)$$

Now, it's just a matter of plugging in the data. :)

## 1. What is the "Trigonometry lighthouse boat problem"?

The "Trigonometry lighthouse boat problem" is a mathematical problem that involves finding the distance between a boat and a lighthouse, given the angle of elevation from the boat to the top of the lighthouse and the height of the lighthouse.

## 2. What are the key concepts in solving the "Trigonometry lighthouse boat problem"?

The key concepts in solving the "Trigonometry lighthouse boat problem" are trigonometric functions (such as sine, cosine, and tangent), right triangle properties, and the Pythagorean theorem.

## 3. How can the "Trigonometry lighthouse boat problem" be applied in real life?

The "Trigonometry lighthouse boat problem" can be applied in real life situations such as navigation, surveying, and architecture. It can also be used to determine the height of tall objects, such as buildings or trees, by measuring the angle of elevation from a known distance.

## 4. What are some tips for solving the "Trigonometry lighthouse boat problem"?

Some tips for solving the "Trigonometry lighthouse boat problem" include drawing a diagram to visualize the problem, labeling the sides and angles of the triangle, and using trigonometric ratios to set up and solve equations.

## 5. Are there any resources available for practicing and improving skills in solving the "Trigonometry lighthouse boat problem"?

Yes, there are many online resources and practice problems available for improving skills in solving the "Trigonometry lighthouse boat problem". Some recommended resources include Khan Academy, MathIsFun, and Mathwarehouse.

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