Right Triangle Trigonometry M 140 Precalculus V. J. Motto

 hypotenuse c b leg SINE COSINE  TANGENT leg a In 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 We’ll label them a, b, and c and the angles  and . Trigonometric functions are defined by taking the ratios of sides of a right triangle.  adjacent opposite hypotenuse c First let’s look at the three basic functions. b leg SINE COSINE  TANGENT leg a They are abbreviated using their first 3 letters

INE OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE We could ask for the trig functions of the angle  by using the definitions. You MUST get them memorized. Here is a mnemonic to help you.  c The sacred Jedi word: b opposite SOHCAHTOA SOHCAHTOA  adjacent INE a OSINE ANGENT PPOSITE DJACENT YPOTENUSE PPOSITE DJACENT YPOTENUSE

It is important to note WHICH angle you are talking about when you find the value of the trig function.  Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so hypotenuse 5 c b 4 opposite opposite  adjacent a 3 Let's choose: sin  = Use a mnemonic and figure out which sides of the triangle you need for tangent. Use a mnemonic and figure out which sides of the triangle you need for sine. tan  =

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. Oh, I'm acute! This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.  5 4 So am I!  3

There are three more trig functions 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.

sec  =  so a h c 5 b 4 o cot  =  a a 3 so Let's try one: sec  = Which trig function is this the reciprocal of?  so a h c 5 b 4 o cot  =  a a 3 so As 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).

TRIGONMETRIC IDENTITIES Trig identities are equations that are true for all angles in the domain. We'll be learning lots of them and use them to help us solve trig equations. RECIPROCAL IDENTITIES These are based on what we just learned. 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). Try this same thing with and what do you get? Which trig function is this?

QUOTIENT IDENTITIES c2 c2 c2  a b c h h o a These are based on what we just learned. Now to discover my favorite trig identity, let's start with a right triangle and the Pythagorean Theorem. a b c  Rewrite trading terms places Divide all terms by c2 c2 c2 c2 h h Move the exponents to the outside o a This one is sin This one is cos Look at the triangle and the angle  and determine which trig function these are.

1 1 cos2 cos2 cos2 sin2 sin2 sin2 This is a short-hand way you can write trig functions that are squared Now to find the two more identities from this famous and often used one. Divide all terms by cos2 cos2 cos2 cos2 What trig function is this squared? 1 What trig function is this squared? Divide all terms by sin2 sin2 sin2 sin2 These three are sometimes called the Pythagorean Identities since they come from the Pythagorean Theorem 1 What trig function is this squared? What trig function is this squared?

RECIPROCAL IDENTITIES QUOTIENT IDENTITIES PYTHAGOREAN IDENTITIES 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.

If 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. Reciprocal of sine so "flip" sine over Sine is the ratio of which sides of a right triangle? When you know 2 sides of a right triangle you can always find the 3rd with the Pythagorean theorem. Now find the other trig functions  3 a "flipped" cos 1 "flipped" tan Draw a right triangle and label  and the sides you know.

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 sin Now use my favourite trig identity Sub in the value of sine that you know Solve this for cos  This matches the answer we got with the other method square root both sides We won't worry about  because angle not negative You can easily find sec by taking reciprocal of cos.

Let's list what we have so far: We need to get tangent using fundamental identities. Simplify by inverting and multiplying Finally 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 1 1. 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 2 Use fundamental trig identities to relate what you know with what you want to find subbing in values you know.

Two angles whose sum is 90° are called complementary angles. The sum of all of the angles in a triangle always is 180° What is the sum of  + ? 90° 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.  c opposite  adjacent to  b  a Since  and  are complementary angles and sin  = cos , sine and cosine are called cofunctions. adjacent to  opposite  This is where we get the name cosine, a cofunction of sine.

Let's see if this is right. Does sec  = cosec ? Looking at the names of the other trig functions can you guess which ones are cofunctions of each other? secant and cosecant tangent and cotangent Let's see if this is right. Does sec  = cosec ? hypotenuse over adjacent hypotenuse over opposite  c opposite  adjacent to  b  a This whole idea of the relationship between cofunctions can be stated as: adjacent to  opposite  Cofunctions of complementary angles are equal.

Cofunctions of complementary angles are equal. cos 27° = sin(90° - 27°) = sin 63° Using the theorem above, what trig function of what angle does this equal? Let's try one in radians. What trig functions of what angle does this equal? The sum of complementary angles in radians is since 90° is the same as Basically any trig function then equals 90° minus or minus its cofunction.

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