Exercises and problems solved using trigonometry

Let's remember that the formula to calculate an angle from radians to degrees is

 

 

where is the angle in radians. Therefore, the angles in degrees are:

 

a

 

Here we have . Therefore, the degrees are

 

 

To obtain the minutes, we multiply the decimal part by 60:

 

 

To obtain the seconds, we multiply the decimal part again by 60:

 

 

Therefore, the angle in sexagesimal form is

 

b

 

Just like in the previous exercise, we use the formula

 

 

c

 

Here we also use the same formula:

 

The formula to convert from degrees to radians is very similar to the previous one

 

 

Thus, the angles are:

 

a

 

We use the formula

 

 

Therefore, the angle measures radians.

 

b

 

We use the formula

 

 

Therefore, the angle measures radians.

 

c

 

We use the formula

 

 

This cannot be simplified since 127 is prime. Therefore, the angle measures .

Let's remember that the formula to calculate an angle from radians to degrees is

 

 

where is the angle in radians. Therefore, the angles in degrees are:

 

a

 

Here we have . Therefore, the degrees are

 

 

b

 

Just like in the previous exercise, we use the formula

 

 

c

 

Here we also use the same formula:

 

The formula to convert from degrees to radians is very similar to the previous one

 

 

Thus, the angles are:

 

a

 

We use the formula

 

 

Therefore, the angle measures radians.

 

b

 

We use the formula

 

 

Therefore, the angle measures radians.

 

c

 

We use the formula

 

 

Therefore, the angle measures .

Let's remember that the formula to calculate an angle from radians to degrees is

 

 

where is the angle in radians. Therefore, the angles in degrees are:

 

a

 

Here we have . Therefore, the degrees are

 

 

b

 

Just like in the previous exercise, we use the formula

 

 

c

 

Here we also use the same formula:

 

The formula to convert from degrees to radians is very similar to the previous one

 

 

Thus, the angles are:

 

a

 

We use the formula

 

 

Thefore, the angle measures radians.

 

b

 

We use the formula

 

 

Therefore, the angle measures radians.

 

c

 

We use the formula

 

 

Therefore, the angle measures .

We know that cosine is the ratio of the adjacent leg to the hypotenuse, so these are 1 and 4 respectively.

 

Applying the Pythagorean theorem we find the value of the leg opposite to

 

 

Thus,

We know that sine is the ratio of the opposite leg to the hypotenuse, so these are 2 and 3 respectively.

Applying the Pythagorean theorem we find the value of the leg adjacent to 

 

 

Thus,

We know that tangent is the ratio of the opposite leg to the adjacent leg, so these are 3 and 1 respectively.

 

Applying the Pythagorean theorem we find the value of the hypotenuse

 

 

Thus,

First, we know that the angle is located in the fourth quadrant of the coordinate plane.

In this quadrant we have but . Therefore,

 

 

From this it follows that

 

 

Since we now have and , the other identities are simpler.

 

 

Note that both cotangent and cosecant can be rationalized. So it would also be correct if we had

 

 

This is obtained by multiplying the previous results by , thus avoiding radicals in the denominator.

The angle is located in the 3rd quadrant of the coordinate plane. From this we know that and .

 

On the other hand, tangent is related to secant through its Pythagorean identity:

 

 

From which we have

 

 

Since , thus . Therefore,

 

 

From this it follows that

 

 

Note that is also a correct answer (by rationalizing the previous result).

 

We also know that . From this it follows that

 

 

The two remaining identities are calculated very simply:

 

 

and

Let's note, first, that the angle is in radians. Also, we are in the first quadrant of the coordinate plane, so and .

 

On the other hand, secant is related to through its Pythagorean identity:

 

 

Furthermore, since and , thus

 

 

Similarly,

 

 

And since , thus it follows that

 

 

With this, the last two trigonometric identities are very simple to calculate:

 

 

and

Here we will assume we have memorized the sine and cosine for some very common angles (, etc.):

 

a

 

To calculate the sine of the angle, we will use some translation identities. Note that

 

 

 

Similarly,

 

 

Finally

 

 

b

 

Just like in the previous case, we will use some translation identities. Note that

 

 

 

Similarly,

 

 

Here we will assume we have memorized the sine and cosine for some very common angles (, etc.):

 

a

 

To calculate the sine of the angle, we will use some translation identities. Note that

 

 

 

Similarly,

 

 

Finally

 

 

b

 

Just like in the previous case, we will use some translation identities. Note that

 

 

Similarly,

 

 

a

 

First we must find an angle that is between and and that equals . To do this, we divide 2655 by 360 and the remainder will be the angle we're looking for:

 

 

where the remainder is 135. Therefore

 

 

Similarly,

 

 

Finally,

 

 

b

 

This is very similar to the previous case. We first divide 840 by 360 and keep the remainder:

 

 

 

So . Thus:

 

 

 

Finally,

Let's observe the following triangle:

 

 

 

There we can see the data we're missing (sides , and angle ). The simplest is angle , since . Therefore

 

 

 

Since the triangle is right-angled, we can use trigonometric functions to calculate the length of the remaining sides. We know that

 

 

so

 

 

Similarly, since , then

 

 

With this we have found all the missing data.

Let's observe the triangle from this exercise:

 

 

 

 

There we can see the data we're missing (sides , and angle ). Just like in the previous case, the simplest is angle , since . Therefore

 

 

Now we don't have the hypotenuse. Therefore we must use tangent to start:

 

 

 

so

 

 

Similarly, since , then

 

 

With this we have found all the missing data.

 

Let's observe the triangle:

 

 

 

The data we're missing are leg and angles and . By the Pythagorean theorem, we know that , so

 

 

Thus, . Moreover,

 

 

So . Finally, since the angles sum to :

 

 

With this we finish solving the triangle.

Note that this triangle is not right-angled. In fact, the triangle is shown in the following figure:

 

 

Where we see that we're missing angles , and side . Since the triangle is not right-angled, we cannot use the Pythagorean theorem, but we can use the law of cosines:

 

 

where we already have all the data. We have

 

 

Therefore . With this, we can now calculate any of the remaining angles using the law of sines:

 

 

from which it follows that

 

 

from which it follows that .

 

Finally,

 

 

with which we solve the entire triangle completely.

Note that the tree (side ) and the shadow (side ) form the following triangle:

 

 

Notice that it's not necessary to calculate side . We're looking for angle , whose tangent is given by

 

 

Using the arctangent, we obtain

 

 

 

Which is the angle we were looking for.
 

Note that between the town and the airship the following triangle is formed:

 

 

where we denote the unknown distance by . We denote the airship's altitude by and the depression angle coincides with angle .

We know that the tangent of is calculated using

 

 

so

 

 

Therefore, the airship must travel 3,763.70 meters, or 2.34 miles.

Let's observe the following figure:

 

 

Notice that a right triangle is formed with points where is the midpoint of the arc.

 

The radius is the hypotenuse of this triangle, the length of is half the chord, that is,

 

 

and angle measures (half the arc). We know that

 

 

 

since is the hypotenuse. Therefore,

 

 

Therefore, the radius measures 21.44 meters (70.3 feet).

There are several ways to solve this problem. We can use Heron's formula or we can try to calculate one of its heights. First let's observe the triangle:

 

 

Where , and .

 

If we draw the height that is perpendicular to , notice that a right triangle is formed where is a leg and is the hypotenuse. Also, notice that the sine of is

 

 

so

 

 

Therefore, the area is

 

Let's observe the following figure, which is a representation of the problem:

 

 

Notice that there are several ways to solve this problem. One is to find the distance by solving triangle ; then we use that distance to find the height.

 

To solve the triangle, notice that angle of triangle is

 

 

therefore, we can now use the law of sines to solve the triangle. However, we first need angle , which is,

 

 

By the law of sines we have

 

 

 

where

 

 

Therefore,

 

 

With this we can now calculate the height of the tree. Notice that triangle is right-angled. Therefore,

 

 

Thus,

 

 

Therefore, the tree is 8.66 meters (28.4 feet) tall.

Look at the following figure of an octagon with its inscribed and circumscribed circles:

 

 

 

Notice that a right triangle is formed between points where is the midpoint of any side of the octagon. Let's observe this right triangle in more detail:

 

 

 

We know that angle . So angle of the right triangle will be . Also, side .

 

 

The two sides of the triangle we're missing are, in fact, the radii of the circles. Starting with side , we have

 

 

 

So

 

 

Therefore, the radius of the inscribed circle is 14.49 meters (47.5 feet).

 

 

Then, side satisfies

 

 

 

so . That is, the radius of the circumscribed circle is 15.68 meters (51.4 feet).
 
 

The figure representing the problem forms a triangle

 

Notice that to calculate side we just need to apply the law of cosines

 

 

 

Performing the operations, we obtain

 

 

 

Which is the distance between cities and that we were looking for.

Notice that the kite string (side ) and the projection of the kite to the ground (side ) form the following right triangle:

 

 

 

Notice that the expression for the sine of angle is:

 

 

Solving for height , we obtain

 

 

 

Which is the height we were looking for.

Notice that the building (side ) and the shadow (side ) form the following right triangle:

 

 

Notice that the expression for the tangent of the angle is given by

 

 

 

Solving for height , we obtain

 

 

 

Which is the height we were looking for.

a

 

We start by writing and with their definition in sines and cosines:

 

 

Then we perform the addition of fractions (with the common denominator):

 

 

 

We notice that , so

 

 

which are the definitions of and . Therefore,

 

 

 

b

 

Here it's convenient to start from the right side of the equation:

 

 

 

We factor out

 

 

Let's remember that the Pythagorean identity for is , so we have

 

 

which was exactly what we wanted to prove.

 

c

 

Here it's also convenient to start from the right side of the identity, factoring :

 

 

We notice that , so

 

 

 

which was what we were looking to prove.

a

 

The simplest way to prove this identity is to start from the left side and write the relationships in terms of sines and cosines:

 

 

since cancels out. This identity could be proven in a single line.

 

 

b

 

We start from the left side and write the relationships in terms of sines and cosines:

 

 

 

Then we add the fractions using the common denominator:

 

 

 

since . Therefore, we arrive at what we wanted to prove.