Trigonometry - Trigonometry

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Question

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 32o. How high is the taller building?

Answer

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.

We are being asked to find the height of the taller building, but this diagram does not provide a triangle that has as one of its sides the entire height of the larger (rightmost and blue) building. However, we can instead find the distance $\overline{\rm HI}$ , and then add that to the 40 foot height of the shorter building to find the entire height of the taller building. Start by finding $\overline{\rm HI}$ :

$\tan32^\circ=\frac{\overline{\rm HI}}{80}$

$80\cdot \tan32^\circ=\overline{\rm HI}$

$80\cdot .62=\overline{\rm HI}$

$49.6=\overline{\rm HI}$

Remember that this is not the full height of the larger building. To find that, we need to add feet. Therefore, the taller building is 104.6 feet tall.

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Question

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?

Answer

This triangle can exist. Since $\measuredangle A$ is a right angle, we can use the Pythagorean Theorem, where is the hypoteneuse:

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Question

While waiting for your sister to finish her bungee jump, you decide to figure out how tall the platform she is jumping off is. You are standing feet from the base of the platform, and the angle of elevation from your position to the top of the platform is degrees. How many feet tall is the platform?

Answer

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.

$\tan 62=\frac{x}{200}$

Make sure to round to places after the decimal.

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Question

When the angle of elevation of the sun is degrees, a flagpole casts a shadow that is feet long. In feet, how tall is the flagpole?

Answer

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.

$\tan 32=\frac{\text{height}}{10}$

$\text{height}=6.25$

Make sure to round to places after the decimal.

The flagpole is feet tall.

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Question

A ladder that is feet long is resting against the side of a house at an angle of degrees. In feet, how far up the side of the house does the ladder reach?

Answer

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.

$\sin 70=\frac{\text{height}}{125}$

$\text{height}=117.46$

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Question

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

Answer

To make sense of the problem, start by drawing a diagram. Label the angle of elevation as 25o, 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 25o. 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:

$\sin25^{\circ}=\frac{10}{w}$

$w=\frac{10}{\sin25^{\circ}}$

$w=\frac{10}{.42}$ (Use a calculator in degree mode to find that $\sin25^\circ=.42$ after rounding to two decimal places)

meters

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

$\frac{w}{10}=\csc25^{\circ}$

$w=10\cdot \csc25^\circ$

$w=10\cdot 2.37$ (Use a calculator in degree mode to find that $\csc25^\circ=2.37$ after rounding to two decimal places)

meters

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

Please note that the answer choice $w=\sqrt{b^2+100}$ is correct based on the Pythagorean Theorem, but does not use all of the provided info to find an exact solution rounded to two decimal places.

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Question

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

Answer

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.

Next, think about which trig functions relate our known angle, 22o, to the base (or adjacent) and the opposite sides of the triangle. If you thought tangent (or cotangent), you are correct! We know that $\tan x =\frac{opposite}{adjacent}$ and $\cot x =\frac{adjacent}{opposite}$ . For simplicity's sake, we'll use tangent to solve this problem. We have:

$\tan 22^\circ=\frac{60}{b}$

$b=\frac{60}{\tan 22^\circ}$

$b=\frac{60}{.4}$ (Use a calculator and round to two places to find that $\tan 22^\circ=.40$ )

meters

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.)

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Question

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

Answer

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 19o 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:

$\sin 19^\circ=\frac{105}{d}$ (where d is the distance between the top of the lighthouse and the boat)

$d=\frac{105}{\sin19^\circ}$

$d=\frac{105}{.33}$ (using a calculator in degree mode and rounding to two digits, we get that $\sin19^\circ=.33$ )

meters

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Question

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 22o to the horizontal, how high above her starting point is she when she arrives at the lookout?

Answer

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 22o. 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:

$\sin22^\circ=\frac{x}{4000}$

$4000\cdot \sin22^\circ=x$

$4000\cdot .37=x$ (using a calculator in degree mode and rounding to two decimals we get that $\sin22^\circ=.37$ )

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

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Question

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 48o. How high is the taller building?

Answer

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.

We are being asked to find the height of the taller building, but this diagram does not provide a triangle that has as one of its sides the entire height of the larger (rightmost and blue) building. However, we can instead find the distance $\overline{\rm HI}$ , and then add that to the 40 foot height of the shorter building to find the entire height of the taller building. Start by finding $\overline{\rm HI}$ :

$\tan48^\circ=\frac{\overline{\rm HI}}{50}$

$50\cdot \tan48^\circ=\overline{\rm HI}$

$50\cdot 1.11=\overline{\rm HI}$

$55.5=\overline{\rm HI}$

Remember that this is not the full height of the larger building. To find that, we need to add feet. Therefore, the taller building is 95.5 feet tall.

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Question

Two buildings with flat roofs are 330 feet apart. The shorter building is 31.4 feet tall. From the roof of the shorter building, the angle of elevation to the edge of the taller building is 61.4o. How high is the taller building?

Answer

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.

We are being asked to find the height of the taller building, but this diagram does not provide a triangle that has as one of its sides the entire height of the larger (rightmost and blue) building. However, we can instead find the distance $\overline{\rm HI}$ , and then add that to the 40 foot height of the shorter building to find the entire height of the taller building. Start by finding $\overline{\rm HI}$ :

$\tan61.4^\circ=\frac{\overline{\rm HI}}{330}$

$330\cdot \tan61.4^\circ=\overline{\rm HI}$

$330\cdot 1.83=\overline{\rm HI}$

$603.9=\overline{\rm HI}$

Remember that this is not the full height of the larger building. To find that, we need to add feet. Therefore, the taller building is 635.3 feet tall.

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Question

You need to build a diagonal support for the bleachers at the local sportsfield. The support needs to reach from the ground to the top of the bleacher. How the support should look is highlighted in blue below. The bleacher wall is 10 feet high and perpendicular to the ground. The owner would like the support to only stick out 3 feet from the bleacher at the bottom. What is the length of the support you need to build?

Answer

It is important to recognize that the bleacher, the ground, and the support form a right triangle with the right angle formed by the intersection of the bleacher wall and the ground. We know the bottom of the support should only be 3ft from the bleacher wall on the ground and the bleacher wall is 10ft high. We will use the Pythagorean Theorem to solve for the length of the support, which is the hypotenuse of this right triangle. Our base of the triangle is 3 feet and the leg is 10 feet.

$a^{2}+b^{2}=c^{2}$

$3^{2}+10^{2}=c^{2}$

$109=c^{2}$

$\sqrt{109}=\sqrt{c^{2}}$

And so we need a support of 10.44 feet long.

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Question

An airplane takes off from point and reaches its maximum altitude of 31,000 ft. The angle from the point of takeoff to the plane at maximum altitude is 55 degrees. What is the distance from point to where the plane reaches maximum altitude?

Answer

We begin by drawing a picture. Let stand for where the plane reaches a maximum altitude of 31,000 ft. We can also assume that if we draw a line straight down from the plane that it will be perpendicular to the ground. This forms a right triangle.

Now we will use our knowledge of right triangles. We know the trigonometric identity, $\sin \left ( \theta \right )=\frac{opposite}{hypotenuse}$ . We can plug 55 in for our $\theta$ and 31,000 in for the opposite side. Solving for the hypotenuse is solving for the distance from the point of takeoff to the plane when it reaches maximum altitude.

$\sin \left ( \theta \right )=\frac{opposite}{hypotenuse}$

$\sin \left ( 55 \right )=\frac{31,000}h{}$

$h=\frac{31,000}{\sin \left ( 55 \right )}$

And so the distance from point to the plane at maximum altitude is 37,844 ft.

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Question

You are reading a billboard perpendicular to the ground walking downtown. You are standing about 500 ft from the base of the billboard and looking up at an angle of 67 degrees. About what height is the part of the billboard you are looking at standing at? Round to the nearest whole number.

Answer

We begin by drawing a picture. Let point be where your eyes are looking on the billboard and point be where you are standing on the ground.

We must solve for the height of the billboard. We have information of the adjacent side of the right triangle formed and we wish to know the height, which will be the opposite side from you. Using trigonometric identities, we will use . We can plug in 500 ft for our adjacent side and 67 degrees for our .

And the height of the section of the billboard you are looking at is 1,178 ft above the ground.

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Question

You have a shadow that measures 66cm from your feet. You are 60cm tall. What is the sun’s angle of elevation? Round to the nearest degree.

Answer

The important thing to note for this problem is that we are working with similar triangles. Your shadow forms a right triangle with its three points being your feet, the top of your head, and the top of your shadow’s head. We have enough information to find the angle of your head’s elevation to your shadow on the ground. Just like your shadow forms a right triangle to your body, a similar right triangle will also be formed to the sun’s altitude. We know that this triangle is similar because it shares the angle opposite your body and it shares a right angle, so it’s third angle must also be equal. Since it shares the angle opposite your body, we must only solve for the angle of your head’s elevation and it will be the same as the sun’s elevation.

We do this by using . We can plug in 60cm for the opposite since that is your height, and 66cm for the adjacent since that is the length of the shadow. We are solving for .

And so the sun is at an elevation of 48 degrees.

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Question

$f(x)=\sin(x)$

Which of the following is equivalent to the function above.

Answer

The cosine graph at zero has a peak at 1.

The first peak in the sine curve is at π/2, so you adjust the cosine with a -π/2 and that gives the answer $\cos(x-\frac{\pi}{2})$

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Question

Which of those below is equivalent to ?

Answer

Think of a right triangle. The two non-right angles are complementary since the angles in a triangle add up to 180. As a result, when we apply the definitions of sine and cosine, we get that the sine of one of these angles is equivalent to the cosine of the other. As a result, the sine of one angle is equal to the cosine of its complement and vice versa.

Therefore, when we look at the angles of a right triangle that includes a 60 degree angle we get

.

Thus the

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Question

Given that $\sin 60^{\circ}= z$ , which of the following must also be true?

Answer

The supplementary angle identity states that in the positive quadrants, if two angles are supplementary (add to 180 degrees), they must have the same sine. In other words,

if $0^{\circ}<\theta<180^{\circ}$ , then $\sin A = \sin(180-A)$

Using this, we can see that

$\sin 60^{\circ} = \sin (180^{\circ}-60^{\circ}) = \sin 120^{\circ}$

Thus, if $\sin 60^{\circ} = z$ , then $\sin 120^{\circ} = z$ also.

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Question

Which of the following is equivalent to ?

Answer

Recall that the cosine and sine of complementary angles are equal.

Thus, we are looking for the complement of 70, which gives us 20.

When we take the cosine of this, we will get an equivalent statement.

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Question

Simplify the following expression:

$\frac{cos(90-x)}{tan(x))}$

Answer

Simplifying this expression relies on an understanding of the cofunction identities. The cofunction identities hinge on the fact that the value of a trig function of a particular angle is equal to the value of the cofunction of the the complement of that angle. In other words,

That means that returning to our initial expression, we can do some substiution.

$\frac{cos(90-x)}{tan(x)}=\frac{sin(x)}{tan(x)}$

We then turn to another identity, namely the fact that tangent is just the quotient of sine and cosine.

$tan(x)=\frac{sin(x)}{cos(x)}$

We can substitute again

$\frac{sin(x)}{tan(x)}=\frac{sin(x)}{\frac{sin(x)}{cos(x)}}$

Yet dividing by a fraction is the same as multipying by the reciprocal.

$\frac{sin(x)}{\frac{sin(x)}{cos(x)}}=sin(x)\cdot\frac{cos(x)}{sin(x)}=cos(x)$

With some cancellation, we have arrived at our answer.

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