Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Right Angle Trigonometry Right Angle Trigonometry What You Will Learn: – 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. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 2 Right Angle Trigonometry Definition: Trigonometry – is the study of the relationships among the angles and sides of a right triangle. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 3 Right Angle Trigonometry Labeling a Triangle Given : Angle B A Hypotenuse A B Opposite Adjacent Side c a b C Opposite Adjacent Side 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 4 Right Angle Trigonometry What makes Trigonometry work? Similar Right Triangles What is required for two right triangles to be similar? 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 5 Right Angle Trigonometry ABC ADE Given a Opposite SideDE Opposite SideBC rightHypotenuse triangle AD = HypotenuseAB AB BC CA = = EA AD DE AB BC = AD DE DE BC = AD AB B D Divide by AB and multiply by DE A E C Opposite SideDE Opposite SideBC HypotenuseAD = HypotenuseAB 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 6 Right Angle Trigonometry Opposite SideDE Opposite SideBC HypotenuseAD = HypotenuseAB ABC ADE AB BC CA = = AD DE EA AB CA = AD EA EA CA = AD AB Adjacent SideEA Adjacent SideCA HypotenuseAD = HypotenuseAB B D Divide by AB and multiply by EA A E C Adjacent SideEA Adjacent SideCA HypotenuseAD = HypotenuseAB 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 7 Right Angle Trigonometry Opposite SideDE Opposite SideBC HypotenuseAD = HypotenuseAB ABC ADE AB BC CA = = AD DE EA BC CA = DE EA BC DE = CA EA Adjacent SideEA Adjacent SideCA HypotenuseAD = HypotenuseAB Opposite SideBC Opposite SideDE = Adjacent Divide bySide BCCAandAdjacent SideEA B D multiply by EA A E C Opposite SideBC Opposite SideDE Adjacent SideCA = Adjacent SideEA 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 8 Right Angle Trigonometry Opposite SideDE Opposite SideBC HypotenuseAD = HypotenuseAB Adjacent SideEA Adjacent SideCA HypotenuseAD = HypotenuseAB Opposite SideBC Opposite SideDE Adjacent SideCA = Adjacent SideEA 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? 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook A B D E C 9 Right Angle Trigonometry sin A = Let’s call cos A = tan A = Opposite SideDE Opposite SideBC HypotenuseAD = HypotenuseAB Adjacent SideEA Adjacent SideCA HypotenuseAD = HypotenuseAB Opposite SideBC Opposite SideDE Adjacent SideCA = Adjacent SideEA So, for every right triangle with an These are the three basic acute angle A, the various ratios of trigonometric functions. 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 A the triangles are similar. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook B D E C 10 Right Angle Trigonometry = Each pairsin ofAequal trigonometric functions cos A = are called co-functions of the acute angles of the tan A = right triangle. sin B = Opposite SideCA HypotenuseAB = cos A cos B = Adjacent SideBC HypotenuseAB = sin A tan B = Opposite SideCA Adjacent SideBC 1 = cot A tan A Opposite SideBC HypotenuseAB Adjacent SideCA HypotenuseAB Opposite SideBC Adjacent SideCA A 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook B C 11 Right Angle Trigonometry Definition of Six Basic Trig Functions 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 A 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook B c a b C 12 Right Angle Trigonometry A mnemonic use to help remember the first three basic trigonometric functions is: SOH-CAH-TOA Sine Opp over Hyp Tangent Opp over Adj over Hyp The cosecant Cosine (csc) is Adj the inverse of the sine. The secant (sec) is the inverse of the cosine. The cotangent (cot) is the inverse of the tangent. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 13 Right Angle Trigonometry What do the graphs of these trigonometric functions look like? 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 14 Right Angle Trigonometry sin The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum value of the sine function? The x-axis scale is –2 to 2. Note that it completes a cycle every 2 radians. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 15 Right Angle Trigonometry sin To find non-baseline divide the baseline the The number of radiansperiods a trig function requires to by complete coefficient. 3. The baseline non-baseline period one cycle isExample: called thesin function’s period. Theis 2/3 or every 120. period cosecant baseline occurs whenThe thebaseline coefficient for foris the 1. The sine’s function is the same. is 0 to 2. baseline period is 2. Its domain 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 16 Right Angle Trigonometry cos The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum value of the cosine function? The x-axis scale is –2 to 2. Note that it completes a cycle every 2 radians. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 17 Right Angle Trigonometry cos The cosine’s baseline period is 2. Cosine domain is 0 to 2. The baseline period for the secant function is the same. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 18 Right Angle Trigonometry tan The y-axis scale is –1.5 to 1.5, but what is the maximum/minimum value of the tangent function? The x-axis scale is –2 to 2. Note that it completes a cycle every radians. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 19 Right Angle Trigonometry tan The tangent’s baseline period is . Tangent domain is –/2 to /2. The baseline period for the cotangent function is the same. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 20 Right Angle Trigonometry csc Note the scale change. The y-axis scale is –5 to 5, but what is the maximum/minimum value of the cosecant function? The x-axis scale is –2 to 2. Note that it completes a cycle every 2 radians. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 21 Right Angle Trigonometry sec The y-axis scale is –10 to 10, but what is the maximum/minimum value of the secant function? The x-axis scale is –2 to 2. Note that it completes a cycle every 2 radians. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 22 Right Angle Trigonometry cot The y-axis scale is –10 to 10, but what is the maximum/minimum value of the cotangent function? The x-axis scale is –2 to 2. Note that it completes a cycle every radians. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 23 Right Angle Trigonometry sin and csc The x-axis scale is –2 to 2. Note that they complete a cycle every 2 radians (right half of graph). 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 24 Right Angle Trigonometry cos and sec The x-axis scale is –2 to 2. Note that they complete a cycle every 2 radians (right half of graph). 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 25 Right Angle Trigonometry 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. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 26 Right Angle Trigonometry Special Triangles Given: Equilateral Triangle AC = AB/2 B AB2 = AC2 + BC2 Let x = AB BC = (3/2) x BC2 = AB2 – AC2 BC2 = AB2 – (AB/2)2 BC2 = AB2 – AB2/ BC2 = (3/4) AB2 BC = (3/2) AB 19 July 2011 4 A C Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook D 27 Right Angle Trigonometry Special Triangles Given: Equilateral Triangle B Let x = AB BC = (3/2) x AC = x/2 2w x These relationships Which relationship are true for any 30you use depends on 60oproblem. triangle the (3/2 ) x A D C x/ 19 July 2011 w3 2 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook w 28 Right Angle Trigonometry For example: given a 30-60 triangle with the hypotenuse of length 10 units, what are the lengths of the other two sides? The largest side is across from which angle? The 30-60 triangle relationship used was: 19 July 2011 30o 10 x x3 53 /2 60o 10/ x = 5 2 /2 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 29 Right Angle Trigonometry Suppose we only knew the short side and its length is 9. What is the length of the other side and the hypotenuse? 30o 2 9 = 18 2x The 30-60 triangle relationship used was: 93 x3 60o x 9 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 30 Right Angle Trigonometry Suppose we only knew the long side and its length is 7. What is the length of the other side and the hypotenuse? x3 = 7 x = 7/3 30o 2 7/3 = 14/3 7 60o 7/ 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 3 31 Right Angle Trigonometry Special Triangles Given: Right Isosceles Triangle AC = BC AB2 = AC2 + BC2 B AB2 = AC2 + AC2 AB2 = x 2 2AC2 x AB = AC2 Let x = AC = BC A Then AB = x 2 19 July 2011 C x Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 32 Right Angle Trigonometry Special Triangles Given: Right Isosceles Triangle Another Form Given AB = x B x x 2 A C x 2 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 33 Right Angle Trigonometry 3 7 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook B 34 Right Angle Trigonometry 3 7 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook B 35 Right Angle Trigonometry 3 7 B The answer is D. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 36 Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 37 Right Angle Trigonometry What You Have Learned: – To find values of trigonometric functions for acute angles, and – To solve problems involving right triangles. 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 38 Right Angle Trigonometry END OF LINE 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 39 Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 40 Right Angle Trigonometry 19 July 2011 Alg2_13_01_RightAngleTrig.ppt Copyrighted © by T. Darrel Westbrook 41 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