Right Angle Trigonometry

19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 2 – To find values of the six trigonometric functions for acute angles, – To understand the two Special Trigonometric triangles, and – To solve problems involving right triangles. What You Will Learn:

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 3 Definition: Trigonometry – is the study of the relationships among the angles and sides of a right triangle.

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 4 Given : Angle AGiven : Angle B Opposite Side Adjacent Side Labeling a Triangle A B C a c b Opposite Side Adjacent Side Hypotenuse

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 5 What makes Trigonometry work? Similar Right Triangles What is required for two right triangles to be similar?

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 6 AB AD BC DE CA EA = AB AD BC DE CA EA = A B C Given a right triangle D E  ABC   ADE AB AD BC DE = AD BC AB = Opposite Side DE Hypotenuse AD Opposite Side BC Hypotenuse AB = Opposite Side DE Hypotenuse AD = Opposite Side BC Hypotenuse AB Divide by AB and multiply by DE

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 7 A B C D E  ABC   ADE AB AD BC DE CA EA = AB AD CA EA = AD CA AB = Opposite Side DE Hypotenuse AD = Opposite Side BC Hypotenuse AB Adjacent Side EA Hypotenuse AD Adjacent Side CA Hypotenuse AB = Adjacent Side EA Hypotenuse AD = Adjacent Side CA Hypotenuse AB Divide by AB and multiply by EA

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 8 A B C D E  ABC   ADE AB AD BC DE CA EA = BC DE CA EA = BC CA DE EA = Opposite Side DE Hypotenuse AD = Opposite Side BC Hypotenuse AB Opposite Side BC Adjacent Side CA Opposite Side DE Adjacent Side EA = Hypotenuse AD = Adjacent Side CA Hypotenuse AB Opposite Side BC Adjacent Side CA = Opposite Side DE Adjacent Side EA Divide by BC and multiply by EA

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 9 A B C D E Opposite Side DE Hypotenuse AD = Opposite Side BC Hypotenuse AB Adjacent Side EA Hypotenuse AD = Adjacent Side CA Hypotenuse AB Opposite Side BC Adjacent Side CA = Opposite Side DE Adjacent Side EA No matter the length of the sides of the right triangle, these ratios remain equal for a given acute angle. So, what does this imply?

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 10 So, for every right triangle with an acute angle A, the various ratios of the opposite side, adjacent side, and the hypotenuse are the same, no matter the length of the sides of the triangle, as long as the angles are the same and the triangles are similar. Opposite Side DE Hypotenuse AD = Opposite Side BC Hypotenuse AB Adjacent Side EA Hypotenuse AD = Adjacent Side CA Hypotenuse AB Opposite Side BC Adjacent Side CA = Opposite Side DE Adjacent Side EA sin A = cos A = tan A =   A B C D E Let’s call These are the three basic trigonometric functions.

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 11 Opposite Side BC Hypotenuse AB Adjacent Side CA Hypotenuse AB Opposite Side BC Adjacent Side CA sin A = cos A = tan A = A B C sin B = cos B = tan B = Opposite Side CA Hypotenuse AB Adjacent Side BC Hypotenuse AB Opposite Side CA Adjacent Side BC = cos A = sin A 1 tan A = Each pair of equal trigonometric functions are called co-functions of the acute angles of the right triangle. = cot A

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 12 Definition of Six Basic Trig Functions A B C a c b Given : Angle A sin A = Opposite Side Hypotenuse cos A = Adjacent Side Hypotenuse tan A = Opposite Side Adjacent Side csc A = Hypotenuse Opposite Side sec A = Hypotenuse Adjacent Side cot A = Adjacent Side Opposite Side

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 13 A mnemonic use to help remember the first three basic trigonometric functions is: SOH-CAH-TOA Sine Opp over Hyp Cosine Adj over Hyp Tangent Opp over Adj The cosecant (csc) is the inverse of the sine. The secant (sec) is the inverse of the cosine. The cotangent (cot) is the inverse of the tangent.

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 14 What do the graphs of these trigonometric functions look like?

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 15 sin  The x-axis scale is –2  to 2 . Note that it completes a cycle every 2  radians. The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum value of the sine function?

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 16 To find non-baseline periods divide the baseline by the  coefficient. Example: sin 3 . The non-baseline period is 2  /3 or every 120 . The baseline period for the cosecant function is the same. The number of radians a trig function requires to complete one cycle is called the function’s baseline period. The baseline occurs when the coefficient for  is 1. The sine’s baseline period is 2 . Its domain is 0 to 2 . sin 

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 17 cos  The x-axis scale is –2  to 2 . Note that it completes a cycle every 2  radians. The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum value of the cosine function?

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 18 The cosine’s baseline period is 2 . Cosine domain is 0 to 2 . The baseline period for the secant function is the same. cos 

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 19 tan  The x-axis scale is –2  to 2 . Note that it completes a cycle every  radians. The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum value of the tangent function?

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 20 The tangent’s baseline period is . Tangent domain is –  /2 to  /2. The baseline period for the cotangent function is the same. tan 

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 21 csc  The x-axis scale is –2  to 2 . Note that it completes a cycle every 2  radians. The y-axis scale is –5 to 5, but what is the maximum/minimum value of the cosecant function? Note the scale change.

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 22 sec  The x-axis scale is –2  to 2 . Note that it completes a cycle every 2  radians. The y-axis scale is –10 to 10, but what is the maximum/minimum value of the secant function?

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 23 cot  The x-axis scale is –2  to 2 . Note that it completes a cycle every  radians. The y-axis scale is –10 to 10, but what is the maximum/minimum value of the cotangent function?

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 24 sin  and csc  The x-axis scale is –2  to 2 . Note that they complete a cycle every 2  radians (right half of graph).

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 25 cos  and sec  The x-axis scale is –2  to 2 . Note that they complete a cycle every 2  radians (right half of graph).

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 26 tan  and cot  The x-axis scale is –2  to 2 . Note that they complete a cycle every 2  radians, and they are shifted  /2 radians from each other.

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 27 A B D C Given: Equilateral Triangle Special Triangles AC = AB / 2 AB 2 = AC 2 + BC 2 BC 2 = AB 2 – AC 2 BC 2 = AB 2 – ( AB / 2 ) 2 BC 2 = AB 2 – AB 2 / 4 BC 2 = ( 3 / 4 ) AB 2 BC = (  3 / 2 ) AB Let x = AB BC = (  3 / 2 ) x START

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 28 A B D C Given: Equilateral Triangle Special Triangles Let x = AB BC = (  3 / 2 ) x AC = x / 2 x/2x/2 x (  3 / 2 ) x 2w2w w w3w3 These relationships are true for any o triangle Which relationship you use depends on the problem.

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 29 For example: given a triangle with the hypotenuse of length 10 units, what are the lengths of the other two sides? 10 The largest side is across from which angle? 60 o 30 o 10 / 2 = 5 5353 The triangle relationship used was: x x/2x/2 x3/2x3/2

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 30 Suppose we only knew the short side and its length is 9. What is the length of the other side and the hypotenuse? 9 2  9 = 18 9393 The triangle relationship used was: x x3x3 2x2x 60 o 30 o

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 31 Suppose we only knew the long side and its length is 7. What is the length of the other side and the hypotenuse? 7/37/3 2  7 /  3 = 14 /  3 7 x  3 = 7 x = 7 /  3 60 o 30 o

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 32 Given: Right Isosceles Triangle Special Triangles A B C AC = BC AB 2 = AC 2 + BC 2 AB 2 = AC 2 + AC 2 AB 2 = 2AC 2 AB = AC  2 Let x = AC = BC Then AB = x  2 x x x  2

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 33 Given: Right Isosceles Triangle Special Triangles A B C Another Form Given AB = x x x2x2 x2x2

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 34 B 3 7

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 35 B 3 7

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 36 The answer is D. B 3 7

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 37

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 38 What You Have Learned: – To find values of trigonometric functions for acute angles, and – To solve problems involving right triangles.

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 39 END OF LINE

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Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 42

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 43

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 44

Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 45