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Basic trigonometric functions

basic trigonometry

Word trigonometry comes from a Greek word trigonon – triangle and metron and that means measurement. Trigonometry is a very important branch of mathematics which studies mostly right triangles and relations between lengths of sides and their angles. In history, trigonometry was mostly useful in astronomy, to track the movement of stars and measure their distance, for observing the motion of the Sun, for Egyptians to determine when the Nile will be flooded and so on.

In this lesson, we’ll define unit circle, work through all trigonometric functions, graphs, and their properties, trigonometric identities, equations and inequalities, and their application in real life.

Basic trigonometric functions

First, you have to remember that in this lesson we’ll mainly be talking about right triangles. Remember that right triangle is a triangle that has one right angle. Right triangle has two legs and hypotenuse. Through years of studying right triangles, people discovered many interesting laws that are valid generally for all right triangles. These laws express the bond between angles and sides.

In a right triangle ∆ABC with hypotenuse AB:

triangle abc

Sin(\alpha) is a ratio of opposite leg and hypotenuse \frac{a}{c}

Cos(\alpha) is a ratio of adjacent leg and hypotenuse \frac{b}{c}

Tan(\alpha) is the ratio of opposite and adjacent leg \frac{a}{b}

Cot(\alpha) is the of adjacent and opposite leg \frac{b}{a}

What does this opposite and adjacent even mean? Let’s observe angle \beta. It is enclosed by hypotenuse and leg “a”. That leg “a” is its adjacent side, the remaining side, the one he has no connections with is the opposite side. You can conclude the same for the angle \alpha. It is enclosed by leg “b” and hypotenuse.

This means that the leg “b” is its adjacent side, and that the remaining leg “a” is its opposite side. These rules are very useful if we lack informations about our triangle. If we know any two sides, we can in few steps calculate all angles and remaining side.

trigonometric ratios

Example 1. Calculate all angles and sides if the hypotenuse in a right triangle is equal to 5, and angle to the point A is equal to 30^{\circ}.

triangle with hypotenuse of 5In task like this, it’s always advisable to draw a sketch because it’s not always easy picturing it in your mind, and markings of triangles are different everywhere.

Now that you have everything drawn on your sketch you can start observing what you can and what you can’t do. We have one angle and hypotenuse, if you look at the laws we wrote down up you can see that we can use either sine or cosine. It does not matter in which order you solve it so we’ll first calculate the length of side “a”. Since we know that angle to the point A is equal to 30^{\circ} and that “a” is the opposite side of this angle we’ll use sine.

sin(30^{\circ}) = \frac{a}{c}
sin(30^{\circ}) = \frac{a}{5}

a = 5 * sin(30^{\circ})

Now that we got to this expression we can calculate it. Simply type in your calculator this expression and you’ll get that a = 2.5. Now we’ll calculate leg “b”. Since we know the angle whose leg “b” is it’s adjacent side we can use cosine.

cos(30^{\circ}) = \frac{b}{c}
cos(30^{\circ}) = \frac{b}{c}

b = 5 * cos(30^{\circ})

b = 4.33

Now we have all three sides and we have to calculate angles. Since this is the right triangle, and the second angle is know we can simply subtract the measurement of these two and get \beta.

\beta = 180^{\circ} - 30^{\circ} - 90^{\circ} = 60^{\circ}

Example 2. Calculate all angles and sides in a right triangle if the length of one leg is equal to 3cm and \alpha = 50^{\circ}.

right triangle with one leg 3cm and angle 50 degreesSince in the task isn’t explicitly written which leg is which length, we’ll calculate it as if a = 3 cm. Now we’ll use tangent and cotangent.

Side “a” is opposite, and side “b” is adjacent side for angle α.

tan(\alpha) = \frac{a}{b}
b = \frac{a}{tan(\alpha)}

b = \frac{3}{tan(50^{\circ})}

b = 2.52 cm

Now we have to calculate the length of hypotenuse.

sin(α) = \frac{a}{c}
sin(50^{\circ}) = \frac{3}{c}

c = \frac{3}{sin(50^{\circ})} = 3.92 cm


Trigonometric functions and similarity

Two triangles are similar if all of their corresponding angles have the same measure, and that corresponding sides are proportional.

What does this imply when observed in right triangle? Let’s say we have two similar right triangles ∆ ABC and ∆ DEF.

two similar right triangles

Since our assumption is that these two triangles are similar we can conclude that:

\frac{a}{c} = \frac{d}{f}

If we take a look at what these sides represent we can conclude that since a is the adjacent side of β and d is adjacent side of ɛ, and also c is the hypotenuse of the triangle ABC and f is the hypotenuse of triangle DEF. this leads us to:

sin(\beta) = sin(\varepsilon)

If we examine this, we’ll see that all trigonometric values of every two corresponding angles are the same.


Trigonometric functions application problems

Trigonometry is used to solve problems that include right triangles. Trigonometry’s application is basically everywhere. We’ll show you some basic ideas through the following few examples.

Example 1. Imagine you are 190 centimeters tall and you decide to go for a walk. Your shadow falls under the angle of 60^{\circ}. How long is your shadow?

First thing in problem tasks is the sketch. You can observe this problem as a trigonometric problem. You and your shadow enclose the right angle. You have enough information for you to solve it. You have one leg of a right triangle, one angle and you’re looking for the other leg. This is leading us toward using function tangent.

sketch a right triangle

tan(60^{\circ}) = (your height) / shadow
shadow = your height * tan(60^{\circ})

shadow = 190 * tan(60^{\circ})

shadow = 329 cm


Example 2. You have to climb on a roof of some cabin that is 2 meter high and you put your ladder on the ground under the 300. How long are your ladders?

sketch of a cabin 2 meters high

sin(30^{\circ}) = (height of the cabin) / (length of the ladders)
length of the ladders = \frac{2}{sin(30^{\circ})}

length of the ladders = 4m


Reciprocal trigonometric functions

Other than our basic trigonometric functions – sine, cosine, tangent and cotangent there are a lot more. Some of them are called reciprocal trigonometric functions cosecant and secant.

Cosecant is the reciprocal of the sine function. This means that if


Secant is the reciprocal of the cosine function. This means that if


Example 1. Find values of sine, cosine, tangent, cotangent, secant and cosecant for the given triangle in angle \alpha.

calculation trigonometric functions

sin( \alpha) = \frac{opposite}{hypotenuse} = \frac{7}{8.6} = 0.813

cos( \alpha) = \frac{adjacent}{hypotenuse} = \frac{5}{cos(8.6)} = 0.58

tan( \alpha) = \frac{sin( \alpha)}{cos( \alpha)} = \frac{0.813}{0.58} = 1.4

cot( \alpha) = \frac{cos( \alpha)}{sin( \alpha)} = \frac{1}{tan( \alpha)} = \frac{1}{1.4} = 0.71

csc( \alpha) = \frac{1}{sin( \alpha)} = \frac{1}{0.813} = 1.23

sec( \alpha) = \frac{1}{cos( \alpha)} = \frac{1}{cos(0.58)} = 1.72


Inverse trigonometric functions

Each trigonometric function has its own inverse. Let’s see how it looks:

sin^{-1}(x) = arcsinx -> The arc that has a sine of x

cos^{-1}(x) = arccosx -> The arc that has a cosine of x

tan^{-1}(x) = arctanx -> The arc that has a tangent of x

cot^{-1}(x) = arccotx -> The arc that has a cotangent of x

Arc from arcus (bow or and arc) -> “arcsin(x)” means “the arc that has a sine of x”

Inverse functions “undo” trigonometric functions.

That means that:

sin^{-1}(sin(x)) = x -> This will apply to any trigonometric function

Inverse trigonometric functions have various applications in real life situations. Let’s see how it can work on examples:

Example 1. Bob went fishing. First he traveled, in his boat 5 meters long, to the west, but didn’t find any fishes there, so decided to travel north. When he was at 7 meters distance from the last point, he noticed storm coming, so he had to come back. For the fastest travel back, he’ll have to adjust his boat directly towards the docks. At which angle will Bob set his boat to?

drawing bobs path

If we imagine Bob’s path as a right triangle whose legs we know, we can easily find out under which angle bob needs to set his boat to.
Since we know two legs, we can use tangent.

tan(\alpha) = \frac{5}{7} \rightarrow \alpha = 35,54^{\circ}

Example 2. Angela is walking her dog Polo. Leash is 5 meters long, and Polo is 3 meters away from Angela. Under which angle is Angela holding a leash?

First, let’s draw a sketch:

angle her dog sketch

Angela, the leash and the ground are making a right triangle with hypotenuse whose length is equal to 5, and one arm 3. Since the known arm is opposite from the angle we are looking for, we can use sine function.

sin \alpha = \frac{3}{5} \rightarrow \alpha = 36,87^{\circ}

Basic trigonometric functions worksheets

  Trigonometric ratios in a right triangles (171.3 KiB, 317 hits)

  Trigonometric ratios of given angles (156.2 KiB, 220 hits)

  Trigonometric ratios of inverse function in a right triangle (193.9 KiB, 285 hits)

  Using trigonometric functions to find a missing side of a right triangle (256.1 KiB, 185 hits)