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The First Steps in Learning Trigonometry - Trigonometric Functions

Updated on May 31, 2010

Diagram 1 - Right Angle Triangle

What is Trigonometry?

Trigonometry is the study of triangles, particularly right triangles. It deals with relationships between the sides and angles of the triangles. These relationships are expressed by the functions of sine, cosine and tangent. These functions are also used in describing the motion of waves.

In this article, we will be discussing the basic uses of the trigonometric functions sin, cos and tan.

An introduction of right triangles is found in the article Pythagorean Theorem. Go check it out if you need a bit of a refresher.

To begin with, let us define what sin, cos and tan mean. These three functions are simply ratios of the sides of triangles that help us relate to an angle in the triangle. We'll be using the angle A to compare these.

sin(A) = a / c (sin is the opposite side divided by the hypotenuse)

cos(A) = b / c (cos is the adjacent side divided by the hypotenuse)

tan(A) = a / b (tan is the opposite side divided by the adjacent)

An easy way to remember this is SOHCAHTOA:

SOH, sin = opposite / hypotenuse

CAH, cos = adjacent / hypotenuse

TOA, tan = opposite / adjacent

Note that tan is not an entirely independent definition, it's just used to simplify our math. If you were to divide sin by cos, you would get tan.

sin(A) / cos(A) = tan(A)

(a / c) / (b / c) = tan(A)

a / b = tan(A)

Based on Diagram 1.

In most cases, the notation for the angle is that of the Greek letter theta Θ.

Solving the Trig Problems

The main trick with trigonometry problems:

If you are given 2 pieces of data in a triangle (i.e. if you're given an angle and a side length or two side lengths) you can solve the entire triangle with the trig ratios and Pythagorean Theorem.

The slight exception to this is if you're given two angles - this would basically be giving you one piece of data since if you have one angle you can find out the other (the two complementary angles add to 90 degrees). You can still find the ratios of the side lengths using the angles that you are given and calculating the trigonometric ratios, but they could be any set of numbers as long as they make up that ratio.

Usually, you'll be given either two side lengths or an angle and a side length though, and you'll be asked to solve the rest of the triangle.

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made with Macromedia Flash

Example 1

In the first example, you are given two side lengths of 8 and 15, respectively. We are asked to solve for the angle Θ.What function are we supposed to use to solve for Θ?

With respect to Θ, we are given the opposite and the adjacent sides. The hypotenuse is unknown. The easiest way to tackle this is to use tan (opposite / adjacent). We could also use the Pythagorean Theorem to solve for the hypotenuse and then use sin or cos to solve Θ, but why make it harder on yourself?

Now in this we don't use the tan function, we use the inverse tan function (on calculators, it is denoted by tan^-1). Since tanΘ is used to calculate the ratio of opposite / adjacent using Θ, inverse tan is used to calculate Θ using the ratio of opposite / adjacent.

inverse tan (8/15) = 28.07 degrees

It's as simple as that. Make sure your calculator is not in radians - we will talk about that in a future article.

Notice that inverse cos (15/17) = inverse sin (8/17) = inverse tan (8/15) = 28.07 degrees.

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made with Macromedia Flash

Example 2

In this example, we are given a side length of 12 and Θ, which is equal to 30 degrees. We are asked to find the sides a and b. We know that b is the hypotenuse, a is adjacent to Θ, and the side length of 12 is opposite of the angle Θ.

That's a good indication that we can use the sin function:

sin (30) = 12 / b

Rearrange the equation:

b = 12 / sin (30)

b = 24

Now that we have b, we can solve for a using Pythagoras, or we can use the angle Θ again, this time using the tan ratio.

tan (30) = 12 / a

a = 12 / tan (30)

a = 20.8

You now have the basics to solve trigonometric ratios consisting of sin, cos and tan.

Now go get some practice!


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    • profile image


      4 years ago

      Thank you very much it helped me a lot .it is much easier for me now.

    • profile image


      5 years ago

      Your illustration is really good , i loved it ,thank you for your effort.

      I am a great fan of tri.

    • mrpopo profile imageAUTHOR


      5 years ago from Canada

      I'm glad it helped!

    • profile image


      5 years ago

      Thank you for this!

      I need to discuss origami and trigonometry within my research paper (on origami and math) due tomorrow, and idk what trig is (i'm a young 8th grader who is in Algebra 1 right now)

      This explained everything to me

    • profile image


      5 years ago


    • profile image


      6 years ago

      plesase tell me some tricks for solvin questions related to trigonometric identities

    • shamela profile image


      6 years ago

      Dear mrpopo,

      I found Trigonometry easy and refreshing to learn from your hub.

      You have given simple steps to remember Sine,Cosine and tangent.

      I enjoyed reading your hub.

    • mrpopo profile imageAUTHOR


      6 years ago from Canada

      Hey mel, I know these look hard at first, but keep practicing and you'll get them!

    • profile image


      6 years ago

      i do not get trigonemetric function and they are hard! thanks but this could not help me either. i do not know what i should do!

    • mrpopo profile imageAUTHOR


      8 years ago from Canada

      Heheh, I hear ya wilbury. We still gotta know some of these though - or at least how to use the calculator to solve them...

      Thanks for the comment!

    • wilbury4 profile image


      8 years ago from England I think?

      Memories of school and college.... Before calculators...

    • mrpopo profile imageAUTHOR


      8 years ago from Canada

      Haha, thanks for the comment Lamme! Math can be fun, and as easy as pi!

      (yeah lame pun, I know)

    • Lamme profile image


      8 years ago

      Great hub. I love it! I could do math for fun LOL

    • mrpopo profile imageAUTHOR


      8 years ago from Canada

      Thanks TGS! I hope it helps :)

    • TheGlassSpider profile image


      8 years ago from On The Web

      Whoa!! This is very impressive. Math has never been my strongest suit and trig was always difficult for me, but you've laid this out in a very easy-to-understand, non-threatening manner. Thank you!!

    • mrpopo profile imageAUTHOR


      8 years ago from Canada

      Thanks Dame Scribe! I agree, it is great to be adept in mathematics, and it's a favourite subject of mine as well.

    • Dame Scribe profile image

      Dame Scribe 

      8 years ago from Canada

      Math is a awesome skill to hold and be proud to work through easily. A favorite subject for me, :) Well presented!


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