This tutorial assumes you are familiar with the trigonometric functions and their derivation from the unit circle.

A tutorial on the trigonometric functions can be found here.

A tutorial on the trigonometric functions and the unit circle can be found here.

What is an Identity?

An identity (in mathematics) is something that is true (more precisely: a tautological relationship).

For example, the following is an identity:

1 + 1 = 2

It says that 1 + 1 is exactly the same as 2.

A common (and important) trigonometric identity is:

sin2θ + cos2θ = 1

All this identity says is that what is on the left hand side is identical to what is on the right hand side.

Why Learn / Use Identities?

Identities (in any branch of mathematics) help us to:

• solve
• simplify
• or gain insight into

mathematical problems.

Identities are a lot like synonyms in language. Instead of using the word red to describe the colour of something we could use one of the following: carmine, cherry, copper, crimson, ruby, ruddy, rust, scarlet. In language, synonyms give spice to communication; in mathematics, it is used to look at things in a different way.

For example, if you were solving an trigonometric problem and ended up with the following expression:

cos2θ × tanθ × secθ × cscθ

you might not think you had achieved much. However, by using basic trigonometric identities, you would see (or maybe have to work out) that the entire expression can be reduced to the value of 1.

How Many Trigonometric Identities are There?

There are an infinite number of trigonometric identities. Only a few are useful, the remainder generally ending up as problems and exercises for students to solve.

Reference Image

Reciprocal Identities

These identities derive from the fundamental definitions of the trigonometric functions:

Given a right angle triangle, we know that:

sinθ = A / C

We also know that:

cscθ = C / A

It should be obvious that sinθ = 1 / cscθ because the ratio C / A (cosecant) is the reciprocal of the ratio A / C (sine).

The same reasoning applies to all the reciprocal identities.

Ratio Identities

These identities also derive from the fundamental definitions of trigonometric functions and are very similar to the reciprocal identities, except that they consist of the ratio of two trigonometric functions:

The most important of these is probably tanθ = sinθ / cosθ (highlighted in blue).

We know that

tanθ = A / B     (from a right angle triangle)
tanθ = y / x     (from the unit circle)

We also know that tanθ is the slope of the hypotenuse.

From the unit circle we know that:

sinθ = y
cosθ = x

Using the identities sinθ and cosθ to substitute for y and x, we get the following:

tanθ = y / x = sinθ / cosθ

A similar derivation can be done using the definitions from a right angle triangle:

sinθ = A / C
cosθ = B / C

In order to be able to substitute our identities, we need to multiply both the numerator and denominator by 1 / C

       A   (1 / C)     (A / C)     sinθ
tanθ = - × ———————  =  ———————  =  ————
       B   (1 / C)     (B / C)     cosθ

The same sort of reasoning applies to the derivation of the remaining ratio identities.

sinθ
————  =  1
sinθ

Product Identities

While these can be derived from the fundamental trigonometric identities, it is often easier to use the ratio identities.

Taking cosθ × tanθ = sinθ, we can derive it as follows:

1) Find an identity for tanθ:

       sinθ
tanθ = ————
       cosθ

2) Substitute this identity for tanθ into the identity cosθ × tanθ = sinθ:

       sinθ
cosθ × ————  =  sinθ
       cosθ

3) Cancel the cosθ terms:

       sinθ
cosθ × ————  =  sinθ
       cosθ

The same sort of reasoning applies to the derivation of the remaining product identities.

sinθ × cscθ = 1

Pythagorean Theorem Identities

Because the trigonometric functions are derived from a right angle triangle and because the trigonometric functions make use of the sides of a right angle triangle, they can be used in the Pythagorean Theorem:

A2 + B2 = C2

sin²θ + cos²θ = 1

This is the most common (or most famous) example of a Pythagorean Theorem identity using trigonometric functions.

It can be arranged in many different ways, the three most common would be:

sin2θ + cos2θ = 1
sin2θ = 1 - cos2θ
cos2θ = 1 - sin2θ

Looking at a right angle triangle, we see that it has 3 sides, normally labelled A, B, and C. With C being the hypotenuse.

Using these sides in the Pythagorean Theorem, we get:

A2 + B2 = C2

To get it to look like sin2θ + cos2θ = 1, we see that we need to get C2 to be equal to 1. This can be done by dividing the entire equation by C2:

A2    B2    C2
—  +  —  =  —
C2    C2    C2

Which gives us:

A2    B2
—  +  —  =  1
C2    C2

This can be rewritten as:

A   A     B   B
— × —  +  — × —  =  1
C   C     C   C

We know that A/C = sinθ and B/C = cosθ, so we can substitute them in the equation giving us:

sinθ × sinθ  +  cosθ × cosθ  =  1

Which, when we collect the terms, gives us the familiar:

sin2θ + cos2θ = 1

1 + cot²θ = csc²θ

Just as we obtained sin2θ + cos2θ = 1 by dividing the terms in the Pythagorean Theorem by C2, we can divide all the terms in the Pythagorean Theorem by A2, which gives us:

A2    B2    C2
—  +  —  =  —
A2    A2    A2

By using similar steps as above, we get the identity:

1 + cot2θ = csc2θ

Three more common arrangements would be:

cot2θ = csc2θ - 1
csc2θ = cot2θ + 1
csc2θ - cot2θ = 1

tan²θ + 1 = sec²θ

Just as we obtained sin2θ + cos2θ = 1 by dividing the terms in the Pythagorean Theorem by C2, we can divide all the terms in the Pythagorean Theorem by B2, which gives us:

A2    B2    C2
—  +  —  =  —
B2    B2    B2

By using similar steps as above, we get the identity:

tan2θ + 1 = sec2θ

Three common arrangements would be:

tan2θ = sec2θ - 1
sec2θ = tan2θ + 1
sec2θ - tan2θ = 1

Trigonometric Identity Problems

All of the following are trigonometric identities. Feel free to verify / prove using the identities described in the tutorial:

  csc2θ
—————————  =  cot2θ
tan2θ + 1

(1 - sin2α)(1 + tan2α) = 1

sec2φ - 1
—————————  =  sin2φ
  sec2φ

sinβ + cosβcotβ = cscβ

1 - sin4γ - 2sin2γcos2γ = cos4γ

It should be obvious that you can create as many identities as you like. It should also be obvious that most (all?) of these are not as useful as the fundamental identities described in this tutorial.

sin2θ + cos2β = 1

The post Understanding Basic Trigonometric Identities appeared first on Complete, Concrete, Concise.

Don’t Panic

1. Don’t panic or freak out. No one is trying to trick you or trip you up (although, often times, word problems are poorly worded)
2. Read and understand the problem.
3. Trigonometric problems are always about right angle triangles. So the trick is to try and break the problem down into right angle triangles.
4. Trigonometry is about the ratios of the lengths of two sides. If we know the length of two sides of a right angle triangle we can calculate the ratio between those two sides. Those ratios are given special names: sine, cosine, tangent, cosecant, secant, cotangent. Because of the properties of right angle triangles, if we know one of the angles then we also know the ratios of all the sides.
5. Look for what ratios or partial ratios you can find. Then work from there.
6. Similar triangles are triangles that have the same shape that have the same angles inside. Often times, we don’t solve the problem directly but look for a similar triangle. The triangle may be bigger or smaller, but it has the same shape as the triangle we are studying.

Basic Use of Trigonometric Definitions

Basic Use and Application of Trigonometric Definitions and Properties of a Right Angle Triangle

Calculating the Height of a Tree

Potiphar’s out walking his cat. He notices that he casts a shadow 2 m long. He also notices that a tree in the garden casts a shadow 7 m long. If Potiphar is 1.5 m tall how tall is the tree? What is the angle of elevation of the sun?

This is a basic trigonometric problem involving similar triangles. Potiphar and his shadow form a right angle triangle. The tree and its shadow form another right angle triangle. The rays of the sun (which are invisible) form the hypotenuse of these right angle triangles.

We know that similar triangles have the same shape and the same angles. We know that the angle or slope of the hypotenuse is the same for both the tree-triangle and the Potiphar-triangle (because rays of sunlight are parallel to each other). Since both triangles are similar, we also know that the ratios of their sides must be the same.
We know the lengths of the sides of the Potiphar-triangle and can use them to calculate a ratio:
A / B = 1.5 / 2 = 0.75
Of the tree-triangle, we know the length of one of the sides but not of the other. However, we know that the ratio of the unknown length to the known length (i.e. shadow) must be the same as for the Potiphar-triangle:
a / b = A / B = 0.75
Plugging in the values we know (and removing the A / B for clarity):
a / 7 = 0.75
Rearranging we get:
a = 0.75 × 7 = 5.25
The tree is 5.25 m tall.
To calculate the elevation of the sun, we know the ratio of height to length. In trigonometric terms, this is the definition of tangent.
tanα = opposite ÷ adjacent = A / B = a / b = 0.75
Looking up (or using the inverse key on the calculator) we see that for tanα to equal 0.75, α must be 36.87°.

Before calculator use was wide spread, you would look up the value in a tangent table. You can see a tangent table with a resolution of 1 degree over here.

Calculating the Height of a Pyramid and Elevation of the Sun

Calculating the Length of a Shadow

The Peace Tower in Ottawa, Canada is 92.2 m tall. On July 1st, 2012, at noon, the sun’s angle of elevation was 67.6°. How long was the shadow cast by the Peace Tower?

We know the height of the tower and we know the angle of the sun (relative to the ground).
Unlike the previous two problems, we only have one right angle triangle – the tower-shadow-triangle. We know if we have two similar triangles, the ratios of their sides must be the same.
We know the angle the sun makes with the ground, so we know one of the angles of the triangle (actually, we know two of them – the other is 90° – so, if we need to, we can find the third angle because all angles in a triangle must sum up to 180°).
All right angle triangles with the same angles are similar triangles – it doesn’t matter if we don’t know the lengths of the sides because we know the ratios those sides will have.
We know that the following must be true:
length of the opposite side ÷ length of the adjacent side = tan(α)
Plugging in the values we know, we get:
92.2 / x = tan(67.6°)
Evaluating (looking up) tan(67.6°) we get:
9.2. / x = 2.43
Rearranging we get
x = 92.2 / 2.43
Simplifying we get:
x = 37.94
The length of the shadow is 37.94 m (and it is probably safe to round it up to 38 m).

Calculating the Height of a Cliff

Calculating the Height of a Lighthouse

Calculating the Length of a Field #1

Calculating the Length of a Field #2

The post How to Solve Trigonometric Problems appeared first on Complete, Concrete, Concise.

While I call this advanced, it does not mean harder or more complicated, it just means more abstract. Understanding the trigonometric functions (sine, cosine, tangent) using right angle triangles is simply a special case of trigonometric functions using the unit circle.

I strongly recommend first reading and understanding the article Understanding Sine, Cosine, and Tangent first, because it explains the history and reasoning behind the trigonometric functions.

This is the way trigonometric functions are generally understood and defined in mathematics.

Trigonometric functions were originally developed and understood from the study of right angle triangles. The problem with using right angle triangles is that trigonometric functions can only be defined for angles between 0° and 90°, but not for angles ≤ 0° or ≥90° because no such right angle triangles exist.

The Unit Circle

The Unit Circle is a common tool in mathematics. It is a circle with a radius that is one unit long.

How long is a unit? It doesn’t matter. A unit could be 3cm, or 5 inches, or 27 light years. It doesn’t matter how long the radius is, as long as we agree that whatever the length of the radius we will call that length one unit.

Having a unit circle means that we know the diameter is two units long and the length of the circumference (perimeter) is 2π units long.

The unit circle (almost) always has its centre at (0,0). This is certainly true for the trigonometric functions.

Because the unit circle is centred at (0,0) and has a radius of one unit, we know that it intersects (crosses) the X and Y axes at (1,0), (0,1), (-1,0), and (0,-1).

Circling the Triangle

Trigonometry is about triangles. The word trigonometry comes from the Greek trigonometria (trigono = triangle and metria = measure). Going from a triangle to a circle may not seem obvious:

1) Draw a circle

2) Draw a line from the centre of the circle to the perimeter. You can draw the line at any angle you like, but make it greater than 0° and less than 90° (just for now)

3) This line is one unit long and is the same as the radius of the circle

4) Drop a vertical line from where the radius intersects the perimeter. This line will be perpendicular to the x-axis (i.e. it intersects / crosses the x-axis and forms a 90° angle)

5) The vertical line, along with the x-axis form two sides of a right angle triangle (the two sides that join at 90° – traditionally called the opposite and adjacent sides of the right angle triangle)

6) The radius forms the hypotenuse of the right angle triangle with an angle α relative to the adjacent side (alpha is math-speak for “whatever angle you chose”)

Trigonometric Functions Using the Unit Circle

Looking at the triangle formed inside the unit circle, we see there is enough information to work out the trigonometric ratios:

To calculate sin(α) we divide the length of the opposite side by the length of the hypotenuse.

The length of the opposite side is: |y₁ – y₂| (the absolute value of the difference between y-coordinates). This is either |(y – 0)| or |0 – y| (depending which way we take the difference of the coordinates), which is the same as the y-coordinate where the radius intersects the perimeter.

The length of the hypotenuse is: 1 (because this is a unit circle and the hypotenuse has the same length as the radius which is exactly one unit).

Therefore, sin(α) = opposite / hypotenuse = y / 1, which is the same as y.

By using a unit circle, we can immediately find the sine of an angle by looking at the y-coordinate where the radius intersects the perimeter.

A similar derivation applies for cosine – which is the same as the x-coordinate where the radius intersects the perimeter, i.e. cos(α) = x

A similar derivation shows that tangent is y / x.

Angles Greater than 90° or Less Than 0°

Consider the following radius drawn at an angle greater than 90°:

Can we use the x and y coordinates the way we did for angles between 0° and 90°?

We can draw a vertical line from where the radius intersects the unit circle and create a right angle triangle as before, but will we get the same answers?

Calculating sin(α) we see that sin(α) calculated as opposite / hypotenuse = y – just as we observed before.

Calculating cos(α) we see that cos(α) calculated as adjacent / hypotenuse ≠ x. The calculated value is positive, whereas the value of the x-coordinate is negative.

Calculating tan(α) we see that tan(α) calculated as opposite / adjacent ≠ y / x. Again, the first value is positive whereas the second value is negative.

We have the correct magnitude (numbers), but the wrong sign (positive / negative).

In the original article on trigonometric functions, I said that the tangent is the slope of the line. Clearly, the slope is negative, so using the unit circle gives us the right answer – whereas, the triangle way of calculating the tangent gives the wrong answer.

What Does All This Mean?

At the beginning of this article I said that angles are measured in a counterclockwise direction relative to the x-axis.

It is not possible to measure the angle α of the right angle triangle in a counterclockwise direction relative to the x-axis if the triangle is oriented this way (which is the case with the example above):

If the triangle is flipped, then it is possible to measure the angle in a counterclockwise direction:

The two triangles are mirror images of one another. The one on the left is what results with the angle is between 90° and 180° on a unit circle. The one on the right is a mirror image and occurs when we multiply the x-coordinates of the left triangle by -1:

When using the traditional (triangle) way of calculating trigonometric relations we (implicitly) orient the triangle so it is like this:

When we get a negative value, it tells us that the triangle is oriented in a different way.

Summary

1. The unit circle allows us to define trigonometric values for all angles – not just those between 0° and 90°
2. Using the unit circle allows us to correctly determine the slope (tangent) of a line
3. Sine and cosine can be read directly off the unit circle

Function	How to Compute	Comment
Sine	y	y-coordinate where the radius intersects the perimeter of the circle
Cosine	x	x-coordinate where the radius intersects the perimeter of the circle
Tangent	y / x	ratio of the y-coordinate to the x-coordinate where the radius intersects the perimeter of the circle. This is the same as the slope of the line.
Cosecant	1 / y	inverse of the y-coordinate where the radius intersects the perimeter of the circle
Secant	1/ x	inverse of the x-coordinate where the radius intersects the perimeter of the circle
Cotangent	x / y	ratio of the x-coordinate to the y-coordinate where the radius intersects the perimeter of the circle

The post Understanding Trigonometric Functions using the Unit Circle (Advanced) appeared first on Complete, Concrete, Concise.

]]> https://complete-concrete-concise.com/mathematics/understanding-sine-cosine-and-tangent/ Sat, 02 Jun 2012 05:31:19 +0000 http://complete-concrete-concise.com/?p=2164 An article explaining trigonometric functions using the unit circle can be found here Using the unit circle is the standard way trigonometric functions are defined and understood in mathematics. I recommend reading and understanding this article first. Later, if you want to understand how trigonometric functions are defined for values greater than 90° or less […]

The post Understanding Sine, Cosine and Tangent appeared first on Complete, Concrete, Concise.

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An article explaining trigonometric functions using the unit circle can be found here

Using the unit circle is the standard way trigonometric functions are defined and understood in mathematics.

I recommend reading and understanding this article first. Later, if you want to understand how trigonometric functions are defined for values greater than 90° or less than 0°, go and read the other article.

Sine is often introduced as follows:



Which is accurate, but causes most people’s eyes to glaze over.

The problem is that from the time humans starting studying triangles until the time humans developed the concept of trigonometric functions (sine, cosine, tangent, secant, cosecant and cotangent) was over 3000 years.

A Little History

The ancients studied triangles. One of the things they did was to compare the lengths of the sides of triangles:



A triangle has three sides so there are 6 different ways to compare sides:

A to B, A to C, B to C, B to A, C to A and C to B

Normally, we would write these as fractions:

A   A   B   B   C   C
-,  -,  -,  -,  -,  -
B   C   C   A   A   B

What they discovered was that if two triangles have the same ratios for their three sides, then the triangles are the same shape – they have the same angles inside even though the size of the triangles may be different.



This turned out to be very, very useful because it meant that if you could find a smaller triangle that had the same shape as a larger triangle, then you could study the smaller triangle and find out things about the big triangle. It works the other way around as well, if you have a very small triangle and can find a larger, easier to work with triangle, then you can study the larger triangle and learn things about the smaller triangle. These types of triangles are called similar triangles.

One of the earliest uses of this was shadow reckoning – using shadows to measure things.

Suppose you want to measure the height of a tree, how would you do it?



You could climb the tree while carrying a length of rope. You would lower the rope until it reached the ground, mark the length of the rope and then measure the length of the rope (once you climbed down). Unfortunately this is (1) slow, (2) dangerous (there is always the risk of falling), and (3) possibly inaccurate since it is not always possible to climb to the very top of a tree (trees trunks and branches get thinner near the top and they may not be strong enough to support you climbing all the way to the top).

You could cut the tree down and then measure it. But if you want to keep the tree, cutting it down is not an option (it is also pretty slow).

What if you wanted to measure the height of a building or a cliff? Using a rope would work (assuming the rope is long enough), but cutting it down wouldn’t.

However, the ancients noticed that the tree (or building, or cliff) forms one side of a right angle triangle (we’ll assume the tree is growing vertically instead of at an angle), its shadow forms the base of a right angle triangle and the rays of sunlight form the hypotenuse.

If we can make a smaller right angle triangle with the hypotenuse at the same angle as the rays of sunlight, then we have a similar triangle. We can then measure the smaller (and similar) triangle and apply what we know about it to the larger triangle:



Because the large and small triangle are similar, it means the ratios of the sides are the same. If we have a stick of a known length and place it perpendicular to the ground (this is side a of the small triangle), it will cast a shadow and we can measure the length of that shadow (this is sideb of the small triangle). We can easily measure the length of the shadow cast by the tree (this is side B of the large triangle. Because the two triangles are similar, we know that the ratio a/b must be the same as the ratio A/B.

NOTE: it doesn’t matter if the ratio is calculated as a/b or b/a – as long as it is calculated the same way for both triangles.

If our measuring stick is 2m tall and it casts a shadow 1.25m long, the ratio a/b is 1.6. Because the triangles are similar, we know that the ratio A/B must be 1.6. If the tree casts a shadow that is 15m long we can plug that number into the equation and write A/15 = 1.6. Multiplying both sides by 15 gives us: (A/15)*15 = 1.6*15. Multiplying through we get A = 24, so the height of the tree is 24m.

Definitely try this at home. Calculate the height of your house by measuring the length of its shadow and relating that to the length of a known measure (a yardstick (or meter stick) is good – if you don’t have a yardstick or meter stick, you can always use a 12 inch (30cm) ruler). Or measure the heights of trees in your neighbourhood.

Thales of Miletus used this technique to measure the height of the Great Pyramid (which was already some 2000 years old by the time Thales measured its height):



Thales waited until the shadow was inline with one face of the pyramid and then measured the length of the shadow plus half the length of the base to get the length of side B of the similar triangle. Obviously he had a measuring stick whose shadow he also measured.

Eventually, somebody realized that instead of always measuring the sides of a triangle and computing the ratios you could create a lookup table which had the ratios for right-angle triangles with various angles.

We know triangles are similar if:

1. the ratios of the 3 sides are the same or
2. all the angles are the same or
3. two sides have the same ratio and one angle is the same (as long as it is not between the two sides)

We can tell if two right-angle triangles are similar by simply measuring one angle – as long as it is not the square (90°) angle. Because all the angles in a triangle must add up to 180° we can easily find the 3^rd angle (180° – 90° – measured angle = remaining angle)

Any other lookup table would require we measure at least two sides and calculate their ratio.

The earliest indication we have of someone doing this is Hipparchus of Nicaea who, in the 2^nd century BCE, compiled a table of trigonometric ratios (these were chords and are related to, but not the same as our trigonometric ratios). The mathematics of the Greeks ended up in India and was worked upon by Indian mathematicians, their work was then taken and worked further by the Arabs who (by the 9^th century) had developed the modern notion of the six trigonometric functions we use and had lookup tables for right-angle triangles with different angles. Around the 12^th or 13^th century the work of the Arabs arrived back in Europe and was translated from Arabic into Latin, the term sinus (from which we get sine) was the translation used for the Arabic word jiba – both of which mean fold. Had the scholars not translated the word, we might be using jiba or jib or something similar instead.

You can read more about the history of trigonometric functions here, here, and here. [NOTE: these are external links and were deemed to be working, safe, and on topic at the time of this article. While I double check links from time to time, I make no guarantee about them. I also appreciate being told if the links are not working or no longer valid.]

Sine

When we ask, “What is the sine of 30°?” the answer doesn’t really have much to do with the angle. The angle describes which right-angle triangle we are talking about and which two sides of the triangle we are interested in – but the answer is the ratio of the lengths of those two sides.

A better way to ask the question would be:

“Given a right-angle triangle that has an angle of 30°, what number do we get when we divide the length of the side furthest away from the angle by the length of the hypotenuse?”

Saying sin(30), is a shorter way of expressing that previous (long) sentence:



Things to remember:

• the value of sine is always between 0 and 1 (more accurately, it is between -1 and 1)
• one of the sides is always the hypotenuse (the longest side) and it is always the denominator
• the opposite side can be sliced by a line from the angle in question (Sine and Sliced both begin with the letter S)

Cosine

When we ask, “What is the cosine of 30°?” the answer doesn’t really have much to do with the angle. The angle describes which right-angle triangle we are talking about and which two sides of the triangle we are interested in – and the answer is the ratio of the lengths of those two sides.

A better way to ask the question would be:

“Given a right-angle triangle that has an angle of 30°, what number do we get when we divide the length of the side closest to the angle by the length of the hypotenuse?”

Saying cos(30), is a shorter way of expressing that previous (long) sentence:



Things to remember:

• the value of cosine is always between 0 and 1 (more accurately, it is between -1 and 1)
• one of the sides is always the hypotenuse (the longest side) and it is always the denominator
• the adjacent side is the one that is connected to the angle (Cosine and Connected both begin with Co)

Tangent

When we ask, “What is the tangent of 30°?” the answer doesn’t really have much to do with the angle. The angle describes which right-angle triangle we are talking about and which two sides of the triangle we are interested in – and the answer is the ratio of the lengths of those two sides.

A better way to ask the question would be:

“Given a right-angle triangle that has an angle of 30°, what number do we get when we divide the length of the side furthest away from the angle by the length of the side closest to the angle?”

Saying tan(30), is a shorter way of expressing that previous (long) sentence:



NOTE: the tangent is the same as the slope of the hypotenuse.

This is important to know / remember when learning Calculus and you hear about the tangent line to a point on a curve. In fact, anytime you hear “tangent line” it means the “slope of the line”

Things to remember:

• the value of tangent is always between 0 and undefined (sometimes, inaccurately, called infinite ∞ – it is more accurate to say “the slope tends to infinity”). As for sine and cosine, it can be positive or negative.
• the hypotenuse is not used in calculating the tangent
• the tangent is the same as the slope of the hypotenuse
• slope is always rise divided by run (height divided by length)
• the run (length) is the side connected to the angle (if you were standing where the angle was, you would be standing on the side)
• the rise (height) is the side furthest from the angle (if you were standing where the angle was and looked at the side, you would see it rising up)

Secant, Cosecant, and Cotangent

Secant, cosecant, and cotangent are the other three trigonometric functions and they are the inverse of the first three trigonometric functions.

Cotangent

This one is easy, it is the inverse of tangent. Instead of calculating the slope of the hypotenuse, we calculate the inverse of the slope.

cot α = adjacent / opposite (or run / rise)

Secant

Secant is NOT the inverse of sine. Secant is the inverse of cosine.

We invert both the way we calculate and the way we name the function.

sec α = hypotenuse / adjacent

Cosecant

Cosecant is NOT the inverse of cosine. Cosecant is the inverse of sine.

We invert both the way we calculate and the way we name the function.

Pay Attention to Detail

Oftentimes, the right angle triangle is drawn this way:



However, the triangle and given angle can be drawn any which way, so it is important to properly identify which sides are which:



The hypotenuse is green.

The adjacent side is blue.

The opposite side is red.

Mnemonic

Nowadays, math seems to focus on teaching “math words” – acronyms or mnemonics that help in remembering the concept. For the trigonometric functions, the following is used: SOH-CAH-TOA (pronounce ‘soh kah toe ah’). It is supposed to help remember the trigonometric function and which sides give which answer.

SOH – Sine = Opposite ÷ Hypotenuse.

CAH – Cosine = Adjacent ÷ Hypotenuse.

TOA – Tangent = Opposite ÷ Adjacent.

To make it easier to remember, the letters are used as the first letters in a whimsical sentence. I created these two because they use homonyms (sound-a-like words) for the trigonometric functions:

Imagine you have just bought something and there is some paperwork to fill out:

Sign Over Here. Co-sign At Home. Take Ownership Anytime.

or

Imagine you are signing up for a vacation and get some friendly advice at the end. This one does have an extra (unused) word at the end, but it has a homonym for each trigonometric function:

Sign Over Here. Co-sign At Home. Tan Only At night.

This article uses modified tree and pyramid images from The Open Clip Art Library. While the artwork is public domain and does not require attribution, I think it is important and polite to acknowledge sources.

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