Basic Trigonometric Functions: Sine, Cosine, and Tangent
Delve into the fundamental trigonometric functions of sine, cosine, and tangent with this introductory lesson. Learn how these functions relate to angles and triangles, and discover their essential properties and applications. Build a strong foundation in trigonometry as you explore the relationships between these functions and their graphical representations.
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Presentation Transcript
LESSON 1: BASIC TRIGONOMETRIC FUNCTIONS SINE, COSINE, AND TANGENT
I) WHAT IS TRIGONOMETRY? Study of the relationship between the angles in a right triangle with the lengths of the sides Basic trigonometry deals mainly with Right Triangles There are three basic trigonometry functions Sine, Cosine, and Tangent [look for them on your calculator] These trig. Functions will give you the ratios of the sides in a right triangle ( ) sin 22.62 = 0.38461538... 13 What do these numbers represent? 5 ( ) cos 22.62 =0.92307692... 22.62 12 ( ) tan 22.62 = 0.41666666...
II) NAMING SIDESOFA RIGHT TRIANGLE When naming the sides of a R.T., they are relative to the angle that you are using H H Opposite Side O Adjacent Side A Hypotenuse Hypotenuse Adjacent Side A O Opposite Side Note: The Adjacent and Opposite side can be switched around depending on which angle you use. Note: The Hypotenuse must be the longest side and opposite from the box When using Trig, make sure you calculator is on DEG mode Deg Degree Mode
III) SOH-CAH-TOA The Trig. Function that you use depends on which sides of a Right Triangle are given SOH-CAH-TOA pposi O A djacent te djacent ypote H A pposite ypote H O = = = T an C os Sin nuse nu s e If you have or want to find either the opposite or Hypotenuse sides, then use SINE If you have or want to find either the adjacent or Hypotenuse sides, then use COSINE If you have or want to find either the opposite or adjacent sides, then use Tangent
Ex: Indicate which sides are given: Opp, Adj, or Hyp Then indicate which trig function should be used to solve the triangle c) b) a) 12 5 5 23 x 6 6 f) e) d) 7 x x 9 x 5
III) FINDING MISSING SIDES The first step is to identify which sides are given, make sure you name the sides based on the angle Using the sides that are given, determine which trig. function to use: SOH - CAH - TOA Write the equation and then use algebra to find the missing side Make sure your calculator is in DEG mode!! ( ) =12 tan 23 x O 12 0.4244748162 =12 23 x A x 12 x = 0.4244748162 28.2702 x =
Ex: Find the length of the missing sides: b) a) c) 58 x x x 4 33 7 55 4 d) e) f) 8.34 x x x 5.3 33 40 55 4
EX: FINDTHEMISSINGSIDESTO 2 DECIMALPLACES USING TANGENT x x 33 7 55 4 55 x ( ) x tan33 = =4 Tan 7 x x 0.6494 =7 4.54 Cross Multiply! 1.4281=4 5.71 = x = x
EX: FINDTHEMISSINGSIDESTO 2 DECIMALPLACES 58 8.34 x x 4 33 ( ) x sin 33 =8.34 ( ) 4 x 4 x x = 4 = cos 58 0.5299 x 0.545=8.34 4.54 Cross Multiply! x = 7.55 0.5299 = = x
EX: FINDTHEMISSINGSIDESTO 2 DECIMALPLACES x x 5.3 40 55 4 ( ) =5.3 sin 40 ( ) 4 x 4 x x = x 4 = cos 55 0.574 5.3 x 5.3 0.642 8.26 0.642 = Cross Multiply! x = 6.97 0.574 = x = x =
IV) USING TRIGTO FIND MISSING ANGLES If you are finding the angles, use the inverse trig functions 1 sin cos tan 1 1 First determine which sides are given and which trig function can be used 12 tan 15 1 1 tan tan tan 15 12 15 38.65980 = Make sure you keep at least 3 to 4 decimal places for your angles = 12 = O 12 = 1 tan 15 A NOTE: You can only Inverse Tan a ratio, it gives you the angle
Ex: Find the degree of the missing angle: 7.8 5 6 5.2 5 5 7 5.7 5 6 4 4
5 6 5.2 5 cos =5.2 tan =5 6 5 5.2 6 0.866 5 5 cos = cos cos 1 tan = tan 1 tan tan ( ) 1 1 1 cos ( ) = = 30 = = 45 1 1
PRACTICE: FINDTHEDEGOFTHEMISSINGANGLE 7.8 7 5 4 ( ) 5 ( ) sin = 4 7 cos = 7.8 sin 5 = 1 cos 1 4 7.8 = 7 = 39.86834 = 55.1501
PRACTICE: FINDTHEMISSINGANGLETOTHE NEARESTDEGREE 5.7 5 6 4 ( ) 6 5 ( ) tan = 4 sin = 5.7 sin tan 1 6 = 4 5 = 1 5.7 = 50.1944 = 44.5679
Q: Given each of the following values below, which of the following can sin not be equal to? = ) sin 0.55 i 3 = ) sin ii 4 3 = ) sin iii 2 = ) sin 0.75 iv = ) sin 1.05 v
KEY IDEAS: The range of sin and cos is between -1 to 1, it can t be bigger than 1 and less than -1 sin ,,cos and tan can all be negative (lesson 2) Opp Hyp Adj Hyp Opp Adj sin cos = = = = sin cos tan + = 2 2 sin cos 1 Pythagorean Thm. Cheong s Line 2 2 Opp Hyp Adj Hyp + = O 2 2 A + 2 2 2 Opp adj Hyp Hyp = = 1 2 2 Hyp
CHALLENGE: TWOBUILDINGARE 70METERSAPART. THE SHORTERBUILDINGIS 50MHIGH. A CABLEISATTACHEDTO BOTHBUILDING. THEANGLEOFINCLINATIONIS 15 . HOW TALLISTHETALLERBUILDING? 15
SIZEOFA TRIANGLE DONT MATTER The tangent ratio is used when you are given the Opposite and Adjacent sides of a R.T. Your calculator must be in Deg mode (Degree) The angle doesn t change when you have a larger similar triangle because the RATIO stays the same Measure the angle: 36.87 12 9 36.87 = tan 0.750 6 When you Tan the angle, it will be equal to the RATIO of the opposite side divided by the adjacent side 3 4 16 12 8
III) COSINE RATIO The Cosine ratio is used when you are given the Adjacent side and Hypotenuse of a R.T. Measure the angle: 36.87 20 15 36.87 = cos 0.800 10 When you Cos the angle, it will be equal to the RATIO of the Adjacent side divided by the Hypotenuse 5 16 12 4 8
