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# Right triangle trigonometry

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### Right triangle trigonometry

1. 1. R I A N G L E
2. 2. hypotenuse leg leg 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 a b c We’ll label them a, b, and c and the angles α and β. Trigonometric functions are defined by taking the ratios of sides of a right triangle. β α First let’s look at the three basic functions. SINE COSINE TANGENT They are abbreviated using their first 3 letters c a == hypotenuse opposite sinα opposite c b == hypotenuse adjacent cosα adjacent b a == adjacent opposite tanα
3. 3. We could ask for the trig functions of the angle β by using the definitions. a b c You MUST get them memorized. Here is a mnemonic to help you. β α The sacred Jedi word: SOHCAHTOA c b == hypotenuse opposite sin β adjacent cos hypotenuse a c β = = opposite tan adjacent b a β = = opposite adjacent SOHCAHTOA
4. 4. It is important to note WHICH angle you are talking about when you find the value of the trig function. a b c α Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so 222 cba =+ Let's choose: 222 543 =+3 4 5 sin α = Use a mnemonic and figure out which sides of the triangle you need for sine. h o 5 3 = opposite hypotenuse tan β = a o 3 4 = opposite adjacent Use a mnemonic and figure out which sides of the triangle you need for tangent. β
5. 5. 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. α This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle. 3 4 5 β Oh, I'm acute! So am I!
6. 6. opposite hypotenuse cosecant = hypotenuse opposite sin = 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. hypotenuse adjacent cos = adjacent opposite tan = adjacent hypotenuse secant = opposite adjacent cotangent = 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.
7. 7. a b c α hypotenuse adjacent iscos 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). 3 4 5 Let's try one: sec α = so 4 3 cot β = 4 5 Which trig function is this the reciprocal of? β adjacent hypotenuse issec ha adjacent opposite istan so opposite adjacent iscot a o
8. 8. 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. θ θ sin 1 cosec = θ θ cos 1 sec = θ θ tan 1 cot = We can discover the quotient identities if we take quotients of sin and cos: == h a h 0 cos sin θ θ Remember to simplify complex fractions you invert and multiply (take the bottom fraction and "flip" it over and multiply to the top fraction). a h h o ⋅ a o = Which trig function is this? θtan= Try this same thing with and what do you get? θ θ sin cos
9. 9. Now to discover my favorite trig identity, let's start with a right triangle and the Pythagorean Theorem. Rewrite trading terms places QUOTIENT IDENTITIES These are based on what we just learned. θ θ θ cos sin tan = θ θ θ sin cos cot = a b c θ 222 cba =+ 222 cab =+ Divide all terms by c2 c2 c2 c2 1 22 =      +      c a c b Move the exponents to the outside Look at the triangle and the angle θ and determine which trig function these o h This one is sin a h This one is cos ( ) ( ) 1cossin 22 =+ θθ
10. 10. 1cossin 22 =+ θθ 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. 1cossin 22 =+ θθ Divide all terms by cos2 θ cos2 θ cos2 θ cos2 θ What trig function is this squared? 1 What trig function is this squared? θθ 22 sec1tan =+ 1cossin 22 =+ θθ Divide all terms by sin2 θ sin2 θ sin2 θ sin2 θ What trig function is this squared? 1 What trig function is this squared? θθ 22 coseccot1 =+ These three are sometimes called the Pythagorean Identities since they come from the Pythagorean Theorem
11. 11. 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 IDENTITIES θ θ sin 1 cosec = θ θ cos 1 sec = θ θ tan 1 cot = QUOTIENT IDENTITIES θ θ θ cos sin tan = θ θ θ sin cos cot = θθ 22 sec1tan =+ θθ 22 coseccot1 =+ PYTHAGOREAN IDENTITIES 1cossin 22 =+ θθ
12. 12. 3 If the angle θ 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 θ and the sides you know. θ When you know 2 sides of a right triangle you can always find the 3rd with the Pythagorean theorem. a 222 31 =+a 228 ==a 22 Now find the other trig functions =θcos h a 22 3=θsec 3 22 = Reciprocal of sine so "flip" sine over =θcosec 3 =θtan a o 22 1 = "flipped" cos =θcot 22 "flipped" tan 3 1 sin =θ h o = 1
13. 13. 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 =θcosec 3 Now use my favourite trig identity 1cossin 22 =+ θθ Sub in the value of sine that you know Solve this for cos θ 9 8 cos2 =θ 3 22 9 8 cos ==θ This matches the answer we got with the other method You can easily find sec by taking reciprocal of cos. We won't worry about ± because angle not negative square root both sides 3 1 sin =θ 1cos 3 1 2 2 =+      θ
14. 14. Let's list what we have so far: =θcosec 3 We need to get tangent using fundamental identities. θ θ θ cos sin tan = Simplify by inverting and multiplying 3 22 cos =θ Finally you can find cot by taking the reciprocal of this answer. 22 3 sec =θ 22 3 3 1 ⋅= 22 1 = 22cot =θ 3 1 sin =θ 3 22 3 1 tan =θ
15. 15. 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 θ 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.
16. 16. The sum of all of the angles in a triangle always is 180° α β a b c What is the sum of α + β? 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. ?sinisWhat α c a ?osisWhat βc c a adjacentto αoppositeβ adjacent to β opposite α Since α and β are complementary angles and sin α = cos β, sine and cosine are called cofunctions. This is where we get the name cosine, a cofunction of sine. 90°
17. 17. Looking at the names of the other trig functions can you guess which ones are cofunctions of each other? α β a b c Let's see if this is right. Does sec α = cosec β? βα cosecsec == b c adjacentto αoppositeβ adjacent to β opposite α secant and cosecant tangent and cotangent hypotenuse over adjacent hypotenuse over opposite This whole idea of the relationship between cofunctions can be stated as: Cofunctions of complementary angles are equal.
18. 18. Cofunctions of complementary angles are equal. cos 27° Using the theorem above, what trig function of what angle does this equal? = sin(90° - 27°) = sin 63° Let's try one in radians. What trig functions of what angle does this equal? 8 tan π       −= 82 cot ππ The sum of complementary angles in radians is since 90° is the same as 2 π 2 π       = 8 3 cot π Basically any trig function then equals 90° minus or minus its cofunction. 2 π
19. 19. ° ° 54sin 36sin 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 ° ° = 36cos 36sin Now that the angles are the same we can use a trig identity to simplify. °= 36tan
20. 20. 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 Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au
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