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Introduction to Trigonometry

Trigonometry (from Greek trigonon "triangle" + metron "measure")

Want to learn Trigonometry? Here is a quick summary. Follow the links for more, or go to Trigonometry Index

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

Right-Angled Triangle

The triangle of most interest is the right-angled triangle . The right angle is shown by the little box in the corner:

Another angle is often labeled θ , and the three sides are then called:

• Adjacent : adjacent (next to) the angle θ
• Opposite : opposite the angle θ
• and the longest side is the Hypotenuse

Why a Right-Angled Triangle?

Why is this triangle so important?

Imagine we can measure along and up but want to know the direct distance and angle:

Trigonometry can find that missing angle and distance.

Or maybe we have a distance and angle and need to "plot the dot" along and up:

Questions like these are common in engineering, computer animation and more.

And trigonometry gives the answers!

Sine, Cosine and Tangent

The main functions in trigonometry are Sine, Cosine and Tangent

They are simply one side of a right-angled triangle divided by another.

For any angle " θ ":

(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan .)

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°) = Opposite Hypotenuse = 2.8 4.9 = 0.57...

The triangle could be larger, smaller or turned around, but that angle will always have that ratio .

Calculators have sin, cos and tan to help us, so let's see how to use them:

Example: How Tall is The Tree?

We can't reach the top of the tree, so we walk away and measure an angle (using a protractor) and distance (using a laser):

• We know the Hypotenuse
• And we want to know the Opposite

Sine is the ratio of Opposite / Hypotenuse :

sin(45°) = Opposite Hypotenuse

Get a calculator, type in "45", then the "sin" key:

sin(45°) = 0.7071...

What does the 0.7071... mean? It is the ratio of the side lengths, so the Opposite is about 0.7071 times as long as the Hypotenuse.

We can now put 0.7071... in place of sin(45°):

0.7071... = Opposite Hypotenuse

And we also know the hypotenuse is 20 :

0.7071... = Opposite 20

To solve, first multiply both sides by 20:

20 × 0.7071... = Opposite

Opposite = 14.14m (to 2 decimals)

The tree is 14.14m tall

Try Sin Cos and Tan

Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.

Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be positive or negative by the rules of Cartesian coordinates , so the sine, cosine and tangent change between positive and negative also.

So trigonometry is also about circles !

Unit Circle

What you just played with is the Unit Circle .

It is a circle with a radius of 1 with its center at 0.

Because the radius is 1, we can directly measure sine, cosine and tangent.

Here we see the sine function being made by the unit circle:

Note: you can see the nice graphs made by sine, cosine and tangent .

Degrees and Radians

Angles can be in Degrees or Radians . Here are some examples:

Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation (see Amplitude, Period, Phase Shift and Frequency ).

When we want to calculate the function for an angle larger than a full rotation of 360° (2 π radians) we subtract as many full rotations as needed to bring it back below 360° (2 π radians):

Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° − 360° = 10°

cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

And when the angle is less than zero, just add full rotations.

Example: what is the sine of −3 radians?

−3 is less than 0 so let us add 2 π radians

−3 + 2 π = −3 + 6.283... = 3.283... rad ians

sin(−3) = sin(3.283...) = −0.141 (to 3 decimal places)

Solving Triangles

Trigonometry is also useful for general triangles, not just right-angled ones .

It helps us in Solving Triangles . "Solving" means finding missing sides and angles.

Example: Find the Missing Angle "C"

Angle C can be found using angles of a triangle add to 180° :

So C = 180° − 76° − 34° = 70°

We can also find missing side lengths. The general rule is:

When we know any 3 of the sides or angles we can find the other 3 (except for the three angles case)

See Solving Triangles for more details.

Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

Trigonometric and Triangle Identities

And as you get better at Trigonometry you can learn these:

Enjoy becoming a triangle (and circle) expert!

Trigonometric Word Problems

In these lessons, examples, and solutions we will learn the trigonometric functions (sine, cosine, tangent) and how to solve word problems using trigonometry.

Related Pages Trigonometry Word Problems Lessons On Trigonometry Inverse trigonometry Trigonometry Worksheets

The following diagram shows how SOHCAHTOA can help you remember how to use sine, cosine, or tangent to find missing angles or missing sides in a trigonometry problem. Scroll down the page for examples and solutions.

How To Solve Trigonometry Problems Or Questions?

Step 1: If no diagram is given, draw one yourself. Step 2: Mark the right angles in the diagram. Step 3: Show the sizes of the other angles and the lengths of any lines that are known. Step 4: Mark the angles or sides you have to calculate. Step 5: Consider whether you need to create right triangles by drawing extra lines. For example, divide an isosceles triangle into two congruent right triangles. Step 6: Decide whether you will need the Pythagorean theorem, sine, cosine or tangent. Step 7: Check that your answer is reasonable. The hypotenuse is the longest side in a right triangle.

How To Use Cosine To Calculate The Side Of A Right Triangle?

Solution: Use the Pythagorean theorem to evaluate the length of PR.

How To Use Tangent To Calculate The Side Of A Triangle?

Calculate the length of the side x, given that tan θ = 0.4

How To Use Sine To Calculate The Side Of A Triangle?

Calculate the length of the side x, given that sin θ = 0.6

How To Solve Word Problems Using Trigonometry?

The following video shows how to use the trigonometric ratio, tangent, to find the height of a balloon.

How To Solve Word Problems Using Sine?

This video shows how to use the trigonometric ratio, sine, to find the elevation gain of a hiker going up a slope.

Example: A hiker is hiking up a 12 degrees slope. If he hikes at a constant rate of 3 mph, how much altitude does he gain in 5 hours of hiking?

How To Use Cosine To Solve A Word Problem?

Example: A ramp is pulled out of the back of truck. There is a 38 degrees angle between the ramp and the pavement. If the distance from the end of the ramp to to the back of the truck is 10 feet. How long is the ramp? Step 1: Find the values of the givens. Step 2: Substitute the values into the cosine ratio. Step 3: Solve for the missing side. Step 4: Write the units

How To Solve Word Problems Using Tangent?

The following video shows how to use trigonometric ratio, tangent, to find the height of a building.

How To Solve Trigonometry Word Problems Using Tangent?

Example: Neil sees a rocket at an angle of elevation of 11 degrees. If Neil is located at 5 miles from the rocket launch pad, how high is the rocket?

How To Determining The Speed Of A Boat Using Trigonometry?

Example: A balloon is hovering 800 ft above a lake. The balloon is observed by the crew of a boat as they look upwards at an angle of 0f 20 degrees. 25 seconds later, the crew had to look at an angle of 65 degrees to see the balloon. How fast was the boat traveling?

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Trigonometry Calculator

Get detailed solutions to your math problems with our trigonometry step-by-step calculator . practice your math skills and learn step by step with our math solver. check out all of our online calculators here .,  example,  solved problems,  difficult problems.

Here, we show you a step-by-step solved example of trigonometry. This solution was automatically generated by our smart calculator:

Starting from the left-hand side (LHS) of the identity

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

 Intermediate steps

Multiplying the fraction by $\sin\left(x\right)$

When multiplying two powers that have the same base ($\sin\left(x\right)$), you can add the exponents

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

Combine fractions with common denominator $\cos\left(x\right)$

Apply the trigonometric identity: $1-\sin\left(\theta \right)^2$$=\cos\left(\theta \right)^2$

Simplify the fraction $\frac{\cos\left(x\right)^2}{\cos\left(x\right)}$ by $\cos\left(x\right)$

Since we have reached the expression of our goal, we have proven the identity

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• Trigonometry

Trigonometry is the study of relations between the side lengths and angles of triangles through the trigonometric functions . It is a fundamental branch of mathematics, and its discovery paved the way towards countless famous results.

In contest math, trigonometry is an integral subfield of both geometry and algebra . Many essential results in geometry are written in terms of the trigonometric functions, such as the Law of Sines and the Law of Cosines ; many more, such as Stewart's Theorem , are most easily proven using trigonometry. In algebra, expressions involving the trigonometric functions appear frequently on contests. These are solved by clever usage of the trigonometric functions' countless identities , which can simplify otherwise unwieldy equations.

Outside of competition math, trigonometry is the backbone of much of analysis. In particular, Fourier Analysis is written almost entirely in the language of the trigonometric functions.

• 1.1 Right triangle definition
• 1.2 Unit circle definition
• 1.3 Taylor series definition
• 2.1 Law of Sines
• 2.2 Law of cosines
• 3 Trigonometric identities

Definitions

The trigonometric functions can be defined in several equivalent ways. The definition usually taught first is the right triangle definition, for its ease of access. An intermediate to olympiad geometry course usually uses the unit circle definition of trigonometry. Beyond the scope of contest math, the Taylor series definition of trigonometry is preferred in order to extend trigonometry to a complex domain.

Right triangle definition

A common mnemonic to remember this is SOH-CAH-TOA , where S ine = O pposite / H ypotenuse, C osine = A djacent / H ypotenuse, and T angent = O pposite / A djacent

More uncommon are the reciprocals of the trigonometric functions, listed below.

Even though it is defined using right triangles, trigonometry is just as useful when used on acute and obtuse triangles. The Law of Sines and Law of Cosines mentioned below generalize the right triangle definition to include all triangles.

Unit circle definition

The benefit of this definition is that it matches the right triangle definition for acute angles, but extends their domain from acute angles to all real-valued angles. As such, this definition is usually preferred in intermediate to olympiad geometry settings.

Taylor series definition

Applications in Geometry

While trigonometry is useful at any level, intermediate competitions are particularly fond of geometric trigonometry questions. In addition to those mentioned

• Law of Sines

Law of cosines

• Trigonometric identities

Trigonometric identities are expressions true for all inputs involving the trigonometric functions. Due to the natural relationship between their definitions, these identities run numerous. In contest math, the most useful of these are:

• Pythagorean identities
• Angle addition identities
• Double angle identities
• Half angle identities
• Sum-to-product identities
• Product-to-sum identities
• Law of Cosines
• Stewart's Theorem

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Trigonometry practice problems

Try solving these as much as you can on your own, and if you need help, look at the hidden solutions. You may use a calculator. You can download a copy of all these questions (Acrobat (PDF) 108kB Jul25 09) to use as you try these on your own. If you are having difficulty, try the Basic Trig Functions sample problems page.

Calculating the length of a side

Length of a path up a hill.

`text{Sin}=\frac{text{Opposite}}{text{Hypotenuse}`

Substituting in the appropriate values,

`text{Sin(12)}=\frac{500}{text{H}}`

`\frac{500}{text{sin(12)}}=text{H}`

sin(12)=0.21 so, `text{H}=\frac{500}{0.21}=2400  m`

The trail up the hill is 2400 m long .

Depth to a bed of coal

`text{Tan}=\frac{text{Opposite}}{text{Adjacent}}`

`text{tan(12)}=\frac{text{O}}{6\ km}`

Rearranging to isolate O,

O = tan(12) * 6 km

Using a calculator, the value of tan(12) is 0.213. So

O = 0.213 * 6 km

So the opposite side of the triangle, the depth of the coal bed, is 1.275 km , or 1275 meters .

Calculating radius of the outer core (seismology)  

`text{Cos}=\frac{text{Adjacent}}{text{Hypotenuse}}`

Substituting in,

`text{cos(52.5)}=\frac{text{A}}{6370\ km}`

Rearranging for A,

A = cos(52.5) * 6370 km so A = 3877 km

So the approximate radius of the core is 3877 km .  Note that this is approximate, because of the bending of the seismic waves as they reflect through the mantle. The actual radius is about 3500 km.

Calculating an angle

Stream gradient.

`2700\ ft\times\frac{1\ mi}{5280\ ft}=0.511\ mi`

(Note - if you need help with this step you can go to the unit conversions page ) Now we can use the formula for sine to calculate the angle x

substituting in,

`text{sin(x)}=\frac{0.511\ mi}{270\ mi}`

sin(x)= 0.00189

to solve for x, the angle, take the inverse sine of each

sin -1 (sin(x)) = sin -1 (0.00189)

since sin -1 (sin(x)) = x , our result is

x = 0.11 degrees

So the slope of the Colorado River is 0.11 degrees.

Angle of Repose

`text{tan(x)}=\frac{11\ cm}{16\ cm}`

tan(x) = .6875

tan -1 (tan(x)) = tan -1 (.6875)

since tan -1 (tan(x)) = x

x = 34.5 degrees , the angle of repose for this sand pile.

Plate tectonics and the angle of subduction

`text{tan(x)}=\frac{100\ km}{200\ km}`

Dividing, we find that

tan(x) = 0.5

tan -1 (tan(x)) = tan -1 (0.5)

x = 26.6 degrees , the angle of subduction.

TAKE THE QUIZ!!  

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Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

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15 Trigonometry Questions And Practice Problems To Do With High Schoolers

Beki Christian

Trigonometry questions address the relationship between the angles of a triangle and the lengths of the sides. By using our knowledge of the rules of trigonometry and trigonometric functions, we can calculate missing angles or sides when we have been given some of the information. 

Here we’ve provided 15 trigonometry questions that will help your students practice the various types of trigonometry questions they will encounter during high school.

Trigonometry in the real world

Trigonometry is used by architects, engineers, astronomers, crime scene investigators, flight engineers and many others.

Trigonometry Quiz

Need to identify the areas of strength and areas for focus in your high school classes? Use this trigonometry check for understanding quiz to understand how best to support your students with trigonometry. Includes topics such as right triangle trigonometry, law of sines, law of cosines and finding area of non-right triangles.

Trigonometry in high school

In trigonometry we learn about the sine function, tangent function, and cosine function. These trig functions are abbreviated as sin, cos, and tan. We can use these to calculate sides and angles in right angled triangles. Later, students will be applying this to a variety of situations as well as learning the exact values of sin, cos, and tan for certain angles.

Students learn about trigonometric ratios: the law of sines, law of cosines, a new formula for the area of a triangle and applying trigonometric theorems to 3D shapes.

Trigonometry for more senior high school students will introduce the reciprocal trig functions, cotangent, secant and cosecant, but you don’t have to worry about these right now!

How to answer trigonometry questions

The way to answer trigonometry questions depends on whether it is a right angled triangle or not.

How to answer trigonometry questions: right angled triangles

If your trigonometry question involves a right angled triangle, you can apply the following relationships, ie SOH, CAH, TOA

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse 

tan θ = opposite/adjacent

The acronym SOH CAH TOA is used so that you can remember which ratio to use.

To answer the trigonometry question:

• Establish that it is a right angled triangle.
• Label the opposite side (opposite the angle) the adjacent side (next to the angle) and the hypotenuse (longest side opposite the right angle).

3. Use the following triangles to help us decide which calculation to do:

How to answer trigonometry questions: non-right triangles

If the triangle is not a right angled triangle then we need to use the sine rule or the cosine rule. 

There is also a formula we can use for the area of a triangle, which does not require us to know the base and height of the triangle.

Sine rule: \frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}

Cosine rule: a^{2}=b^{2}+c^{2}-2bc \cos(A)

Area of a triangle: Area = \frac{1}{2}ab \sin(C)

• Establish that it is not a right angled triangle.
• Label the sides of the triangle using lowercase a, b, c.
• Label the angles of the triangle using upper case A, B and C.
• Opposite sides and angles should use the same letter, for example, angle A is opposite to side a.

Trigonometry questions

In high school geometry, trigonometry questions focus on the understanding of sin, cos, and tan (SOHCAHTOA) to calculate missing sides and angles in right triangles. 

Trigonometry questions: missing side

1. A zip wire runs between two posts, 25m apart. The zip wire is at an angle of 10^{\circ} to the horizontal. Calculate the length of the zip wire.

2. A surveyor wants to know the height of a skyscraper. He places his inclinometer on a tripod 1m from the ground. At a distance of 50m from the skyscraper, he records an angle of elevation of 82^{\circ} .

What is the height of the skyscraper? Give your answer to one decimal place.

Total height = 355.8+1=356.8m.

3. Triangle ABC is isosceles. Calculate the height of triangle ABC.

To solve this we split the triangle into two right angled triangles.

Trigonometry questions: missing angles

4. A builder is constructing a roof. The wood he is using for the sloped section of the roof is 4m long and the peak of the roof needs to be 2m high. What angle should the piece of wood make with the base of the roof?

5. A ladder is leaning against a wall. The ladder is 1.8m long and the bottom of the ladder is 0.5m from the base of the wall. To be considered safe, a ladder must form an angle of between 70^{\circ} and 80^{\circ} with the floor. Is this ladder safe?

Not enough information

Yes it is safe.

6. A helicopter flies 40km east followed by 105km south. On what bearing must the helicopter fly to return home directly?

Since bearings are measured clockwise from North, we need to do 360-21=339^{\circ}.

In geometry, trigonometry questions ask students to solve a variety of problems including multi-step problems and real-life problems. We also need to be familiar with the exact values of the trigonometric functions at certain angles. 

We look at applying trigonometry to 3D problems as well as using the sine rule, cosine rule, and area of a triangle.

Trigonometry questions: SOHCAHTOA

7.  Calculate the size of angle ABC. Give your answer to 3 significant figures.

8. Kevin’s garden is in the shape of an isosceles trapezoid (the sloping sides are equal in length). Kevin wants to buy enough grass seed for his garden. Each box of grass seed covers 15m^2 . How many boxes of grass seed will Kevin need to buy?

To calculate the area of the trapezoid, we first need to find the height. Since it is an isosceles trapezium, it is symmetrical and we can create a right angled triangle with a base of \frac{10-5}{2} .

We can then find the area of the trapezoid:

Number of boxes: 88.215=5.88

Kevin will need 6 boxes.

Trigonometry questions: exact values

9.   Which of these values cannot be the value of \sin(\theta) ?

10. . Write 4sin(60) + 3tan(60) in the form a\sqrt{k}.

Trigonometry questions: 3D trigonometry

11. Work out angle a, between the line AG and the plane ADHE.

We need to begin by finding the length AH by looking at the triangle AEH and using pythagorean theorem.

We can then find angle a by looking at the triangle AGH.

12.   Work out the length of BC.

First we need to find the length DC by looking at triangle CDE.

We can then look at triangle BAC.

Trigonometry questions: sine/cosine rule

13. Ship A sails 40km due West and ship B sails 65km on a bearing of 050^{\circ} . Find the distance between the two ships.

The angle between their two paths is 90+50=140^{\circ} .

\begin{aligned} a^{2}&=b^{2}+c^{2}-2bc \cos(A)\\\\ a^{2}&=40^{2}+65^{2}-2\times 40 \times 65 \cos(140)\\\\ a^{2}&=5825-5200 \cos(140)\\\\ a^{2}&=9808.43\\\\ a&=99.0\mathrm{km} \end{aligned}

14.   Find the size of angle B.

First we need to look at the right angled triangle.

Then we can look at the scalene triangle.

Trigonometry questions – area of a triangle

15. The area of the triangle is 16cm^2 . Find the length of the side x .

\begin{aligned} \text{Area }&=\frac{1}{2}ab \sin(C)\\\\ 16&=\frac{1}{2} \times x \times 2x \times \sin(40)\\\\ 16&=x^{2} \sin(40)\\\\ \frac{1}{\sin(40)}&=x^{2}\\\\ 24.89&=x^{2}\\\\ 5.0&=x \end{aligned}

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The content in this article was originally written by secondary school maths teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Christi Kulesza.

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Trigonometry : Solving Word Problems with Trigonometry

Study concepts, example questions & explanations for trigonometry, all trigonometry resources, example questions, example question #1 : solving word problems with trigonometry.

You can draw the following right triangle using the information given by the question:

Since you want to find the height of the platform, you will need to use tangent.

You can draw the following right triangle from the information given by the question.

In order to find the height of the flagpole, you will need to use tangent.

You can draw the following right triangle from the information given in the question:

In order to find out how far up the ladder goes, you will need to use sine.

In right triangle ABC, where angle A measures 90 degrees, side AB measures 15 and side AC measures 36, what is the length of side BC?

This triangle cannot exist.

Example Question #5 : Solving Word Problems With Trigonometry

A support wire is anchored 10 meters up from the base of a flagpole, and the wire makes a 25 o angle with the ground. How long is the wire, w? Round your answer to two decimal places.

23.81 meters

28.31 meters

21.83 meters

To make sense of the problem, start by drawing a diagram. Label the angle of elevation as 25 o , the height between the ground and where the wire hits the flagpole as 10 meters, and our unknown, the length of the wire, as w. 

Now, we just need to solve for w using the information given in the diagram. We need to ask ourselves which parts of a triangle 10 and w are relative to our known angle of 25 o . 10 is opposite this angle, and w is the hypotenuse. Now, ask yourself which trig function(s) relate opposite and hypotenuse. There are two correct options: sine and cosecant. Using sine is probably the most common, but both options are detailed below.

We know that sine of a given angle is equal to the opposite divided by the hypotenuse, and cosecant of an angle is equal to the hypotenuse divided by the opposite (just the reciprocal of the sine function). Therefore:

To solve this problem instead using the cosecant function, we would get:

The reason that we got 23.7 here and 23.81 above is due to differences in rounding in the middle of the problem. 

Example Question #6 : Solving Word Problems With Trigonometry

When the sun is 22 o above the horizon, how long is the shadow cast by a building that is 60 meters high?

To solve this problem, first set up a diagram that shows all of the info given in the problem. 

Next, we need to interpret which side length corresponds to the shadow of the building, which is what the problem is asking us to find. Is it the hypotenuse, or the base of the triangle? Think about when you look at a shadow. When you see a shadow, you are seeing it on something else, like the ground, the sidewalk, or another object. We see the shadow on the ground, which corresponds to the base of our triangle, so that is what we'll be solving for. We'll call this base b.

Therefore the shadow cast by the building is 150 meters long.

If you got one of the incorrect answers, you may have used sine or cosine instead of tangent, or you may have used the tangent function but inverted the fraction (adjacent over opposite instead of opposite over adjacent.)

Example Question #7 : Solving Word Problems With Trigonometry

From the top of a lighthouse that sits 105 meters above the sea, the angle of depression of a boat is 19 o . How far from the boat is the top of the lighthouse?

423.18 meters

318.18 meters

36.15 meters

110.53 meters

To solve this problem, we need to create a diagram, but in order to create that diagram, we need to understand the vocabulary that is being used in this question. The following diagram clarifies the difference between an angle of depression (an angle that looks downward; relevant to our problem) and the angle of elevation (an angle that looks upward; relevant to other problems, but not this specific one.) Imagine that the top of the blue altitude line is the top of the lighthouse, the green line labelled GroundHorizon is sea level, and point B is where the boat is.

Merging together the given info and this diagram, we know that the angle of depression is 19 o  and and the altitude (blue line) is 105 meters. While the blue line is drawn on the left hand side in the diagram, we can assume is it is the same as the right hand side. Next, we need to think of the trig function that relates the given angle, the given side, and the side we want to solve for. The altitude or blue line is opposite the known angle, and we want to find the distance between the boat (point B) and the top of the lighthouse. That means that we want to determine the length of the hypotenuse, or red line labelled SlantRange. The sine function relates opposite and hypotenuse, so we'll use that here. We get:

Example Question #8 : Solving Word Problems With Trigonometry

Angelina just got a new car, and she wants to ride it to the top of a mountain and visit a lookout point. If she drives 4000 meters along a road that is inclined 22 o to the horizontal, how high above her starting point is she when she arrives at the lookout?

9.37 meters

1480 meters

3708.74 meters

10677.87 meters

1616.1 meters

As with other trig problems, begin with a sketch of a diagram of the given and sought after information.

Angelina and her car start at the bottom left of the diagram. The road she is driving on is the hypotenuse of our triangle, and the angle of the road relative to flat ground is 22 o . Because we want to find the change in height (also called elevation), we want to determine the difference between her ending and starting heights, which is labelled x in the diagram. Next, consider which trig function relates together an angle and the sides opposite and hypotenuse relative to it; the correct one is sine. Then, set up:

Therefore the change in height between Angelina's starting and ending points is 1480 meters. 

Example Question #9 : Solving Word Problems With Trigonometry

Two buildings with flat roofs are 50 feet apart. The shorter building is 40 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 48 o . How high is the taller building?

To solve this problem, let's start by drawing a diagram of the two buildings, the distance in between them, and the angle between the tops of the two buildings. Then, label in the given lengths and angle. 

Example Question #10 : Solving Word Problems With Trigonometry

Two buildings with flat roofs are 80 feet apart. The shorter building is 55 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 32 o . How high is the taller building?

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