R I A N G L E

hypotenuseleglegIn a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse a b

cWell label them a, b, and c and the angles and . Trigonometric functions are defined by taking the ratios of sides of a right triangle.First lets look at the three basic functions.SINECOSINETANGENTThey are abbreviated using their first 3 letters

We could ask for the trig functions of the angle by using the definitions. a b

cYou MUST get them memorized. Here is a mnemonic to help you.The sacred Jedi word:SOHCAHTOASOHCAHTOA

It is important to note WHICH angle you are talking about when you find the value of the trig function. a b

cLet's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so Let's choose:345sin = Use a mnemonic and figure out which sides of the triangle you need for sine.oppositehypotenusetan = Use a mnemonic and figure out which sides of the triangle you need for tangent.

You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle.This method only applies if you have a right triangle and is only for the acute angles (angles less than 90) in the triangle.345

There are three more trig functions. They are called the reciprocal functions because they are reciprocals of the first three functions.Oh yeah, this means to flip the fraction over.Like the first three trig functions, these are referred to by the first three letters except for cosecant since it's first three letters are the same as for cosine.Best way to remember these is learn which is reciprocal of which and flip them.

a b

cAs a way to help keep them straight I think, The "s" doesn't go with "s" and the "c" doesn't go with "c" so if we want secant, it won't be the one that starts with an "s" so it must be the reciprocal of cosine. (have to just remember that tangent & cotangent go together but this will help you with sine and cosine).345socot = Which trig function is this the reciprocal of?so

We can discover the quotient identities if we take quotients of sin and cos:Remember to simplify complex fractions you invert and multiply (take the bottom fraction and "flip" it over and multiply to the top fraction).Which trig function is this?

Now to discover my favorite trig identity, let's start with a right triangle and the Pythagorean Theorem.Rewrite trading terms placesDivide all terms by c2c2c2c2Move the exponents to the outsideLook at the triangle and the angle and determine which trig function these are.This one is sinThis one is cos

This is a short-hand way you can write trig functions that are squaredNow to find the two more identities from this famous and often used one.Divide all terms by cos2cos2cos2cos2What trig function is this squared?1What trig function is this squared?Divide all terms by sin2sin2sin2sin2What trig function is this squared?1What trig function is this squared?These three are sometimes called the Pythagorean Identities since they come from the Pythagorean Theorem

All of the identities we learned are found in the back page of your book under the heading Trigonometric Identities and then Fundamental Identities.You'll need to have these memorized or be able to derive them for this course.RECIPROCAL IDENTITIESQUOTIENT IDENTITIESPYTHAGOREAN IDENTITIES

3If the angle is acute (less than 90) and you have the value of one of the six trigonometry functions, you can find the other five.Sine is the ratio of which sides of a right triangle?Draw a right triangle and label and the sides you know.When you know 2 sides of a right triangle you can always find the 3rd with the Pythagorean theorem.aNow find the other trig functionsReciprocal of sine so "flip" sine over"flipped" cos"flipped" tan1

There is another method for finding the other 5 trig functions of an acute angle when you know one function. This method is to use fundamental identities. We'd still get cosec by taking reciprocal of sinNow use my favourite trig identitySub in the value of sine that you knowSolve this for cos This matches the answer we got with the other methodYou can easily find sec by taking reciprocal of cos.We won't worry about because angle not negativesquare root both sides

Let's list what we have so far:We need to get tangent using fundamental identities.Simplify by inverting and multiplyingFinally you can find cot by taking the reciprocal of this answer.

SUMMARY OF METHODS FOR FINDING THE REMAINING 5 TRIG FUNCTIONS OF AN ACUTE ANGLE, GIVEN ONE TRIG FUNCTION.METHOD 11. Draw a right triangle labeling and the two sides you know from the given trig function.2. Find the length of the side you don't know by using the Pythagorean Theorem.3. Use the definitions (remembered with a mnemonic) to find other basic trig functions.4. Find reciprocal functions by "flipping" basic trig functions.METHOD 2Use fundamental trig identities to relate what you know with what you want to find subbing in values you know.

The sum of all of the angles in a triangle always is 180 abcWhat is the sum of + ?Since we have a 90 angle, the sum of the other two angles must also be 90 (since the sum of all three is 180).Two angles whose sum is 90 are called complementary angles.adjacent to opposite adjacent to opposite Since and are complementary angles and sin = cos , sine and cosine are called cofunctions.This is where we get the name cosine, a cofunction of sine.90

Looking at the names of the other trig functions can you guess which ones are cofunctions of each other? abcLet's see if this is right. Does sec = cosec ? adjacent to opposite adjacent to opposite secant and cosecanttangent and cotangenthypotenuse over adjacenthypotenuse over oppositeThis whole idea of the relationship between cofunctions can be stated as:Cofunctions of complementary angles are equal.

Cofunctions of complementary angles are equal.cos 27Using the theorem above, what trig function of what angle does this equal?= sin(90 - 27)= sin 63Let's try one in radians. What trig functions of what angle does this equal?

We can't use fundamental identities if the trig functions are of different angles.Use the cofunction theorem to change the denominator to its cofunctionNow that the angles are the same we can use a trig identity to simplify.

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephens School Carramarwww.ststephens.wa.edu.au