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Review of Basic Trigonometry

Review of Basic Trigonometry. A mathematical thinking race!. Get a pencil and paper to write your answers to practice problems as you move through each slide. Repeat as needed until you remember your trig basics.

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Review of Basic Trigonometry

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  1. Review of Basic Trigonometry A mathematical thinking race! • Get a pencil and paper to write your answers to practice problems as you move through each slide. • Repeat as needed until you remember your trig basics. • Look at the links on our class website to find more trig lessons and practice if needed. • Trig is also reviewed in the each section of chapter one of your textbook. You MUST know your trig, to be successful in calculus!

  2. 2. In geometry, you learned that an altitude of an equilateral triangle forms what special triangles? • In geometry, you learned that one diagonal of a square forms what special triangles? Answer: 45 – 45 – 90 or “isosceles right triangles” Answer: 30 – 60 – 90 triangles

  3. 3. How many families of Pythagorean triples can you remember from geometry? Don’t list any multiples in the same family! 4. In geometry, what variables represented the three sides of a 45-45-90 family? Answer: (x, x, x ) x x x 5. In geometry, what variables represented the three sides of a 30-60-90 family? Answer: (x, x , 2x) 2x 2x x x x Answer: (3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17) (9, 40, 41) . . .that’s enough

  4. d 7 14 a 7 c b 30 7 7 e 21 60 7 7 1 g 30 7 h 45 f 1 j 45 k 6. Find the missing side in each triangle. (Click to get answer and then click to get next get triangle.) 14

  5. Answer: SOHCAHTOA sin = cos = tan = • 8. What equation would you use to find the needed angle of elevation if you want to install a 24.2 ft escalator to reach a height of 11.5 ft? Solve your equation. 24.2 Answer: 11.5 xo 7. In geometry, you also learned the right triangle definitions of sin, cos and tan. What are those definitions?

  6. 9. In trigonometry, you learned the definitions of the reciprocals of sin, cos and tan. What are those definitions? Answer: Answer: • 10. Next, instead of degrees, you learned to measure angles and arcs in radians. What is the definition of radian? 11. Radians are easy to use in a UNIT CIRCLE because what is the radius of a unit circle? Answer: 1 12. So what is the circumference (or total arc length) around a UNIT CIRCLE? Answer: 2 13. Radians = arc length/radius, so how many radians are there in one complete revolution around the unit circle? Answer: 2/1 = 2 Answer:  14. A semi-circle or 180o is how many radians?

  7. 15. To convert 57o to radians what would you multiply or divide by? Answer: • To convert radians to degrees, what would you • multiply or divide by? Answer: 17. If  = 180o, then you should recognize common conversions . Convert 30o to radians. Answer: /6 18. Convert 45o to radians. Answer: /4 19. Convert 60o to radians. Answer:  /3 Answer:  /2 20. Convert 90o to radians.

  8. 21. In STANDARD POSITION, we measure angles of rotation from zero radians going counter-clockwise. If there are 2 radians in one entire revolution, then what is the measure of each of the QUADRANT angles shown with colored arcs below? Answer: Blue angle in radians? Answer: /2 Green angle in radians? Answer:  Purple angle in radians? Pink angle in radians? Answer: 2

  9. 22. Do you recognize the FOUR angles that would form 45-45-90 triangles? BUT, now can you give these four angles in RADIANS? (Coterminal angles end up at the same places. We could add or subtract multiples of 2for more revolutions that end up in these four places .) Answer: Answer: Answer: Blue angle in radians? Answer: /4 Green angle in radians? Purple angle in radians? Pink angle in radians?

  10. 23. Now visualize four TALL 30-60-90 triangles in STANDARD POSITION. (Next slide we’ll flip them the SHORT way.) Can you name these four angles in RADIANS? Answer: Answer: Blue angle in radians? Answer: /3 Green angle in radians? Purple angle in radians? Pink angle in radians? Answer:

  11. 24. Now visualize four SHORT 30-60-90 triangles in STANDARD POSITION. Can you name these four angles in RADIANS? Answer: Answer: Answer: Blue angle in radians? Answer: /6 Green angle in radians? Purple angle in radians? Pink angle in radians?

  12. Angles that measure more than 2 are more than one revolution: the red angle below is a 2π revolution plus 5π/4 more, i.e., 13π/4. • NEGATIVE angles are measured CLOCKWISE starting at zero. every additional 2 radians is one more revolution. Give the RADIAN measure of each angle below. Answer: Answer: Blue angle in radians? Answer: - Green angle in radians? What is the measure of a counter-clockwise angle of three complete revolutions that terminates at the same place as zero radians? Answer: 6 Add another counter-clockwise revolution to the red angle?

  13. COTERMINAL angles end up at the same terminal side like • or 0 and 2. Which of these angles is coterminal • to ? Answer: 27. If n represents any integer, which expression below can be used to give all angles coterminal to ? Answer: Recall n = all integers so this covers all positive and negative rotations at same time. 28. Which expression below represents ALL angles that terminate at any of the four quadrant axes? (Let n = any integer) Answer:

  14. 29. Right triangle (SOHCAHTOA) definitions are necessary whenever the hypotenuse is not equal to one, but when the hyp = 1, the UNIT CIRCLE DEFINTIONS simply become the x & y coordinates. sin a = = y (x,y) cos a = = x hyp = 1 y= opp tan a = a In the diagram shown, if a = radians, find each: (x, y) = ( ? , ? ) sin a = csc a = cos a = sec a = tan a = cot a = adj =x

  15. 30. Put it all together! Name the radian measure (in the box), the (x, y) coordinates - which are also the cos & sin – and the tangent at each of the special angles. You need to recognize each in a snap! ( 0 , 1 ) tan = undef cos sin tan = 0 ( -1 , 0 ) ( 1 , 0 ) tan = 0 (0 , -1 ) tan = undef Click to see answers ( , ) tan = tan =( , ) ( , ) tan = tan =( , ) ( , ) tan = ( , ) tan = tan =( , ) tan = ( , ) ( , ) tan = tan =( , ) ( , ) tan = tan =( , ) ( , ) tan = tan =( , ) ( , ) tan = ( , ) tan =

  16. 31. Now practice all six ratios in random order. Visualize a unit circle, but don’t waste time drawing it. Practice until you can do these quickly and confidently! Click to check your answers.

  17. 32. Find the exact value of cos x in the diagram below. (-3, 4) Answer: This makes a 3-4-5 ∆. Since hyp  0, we use SOHCAHTOA definitions for the reference angle inside the triangle. cos x = 5 4 x -3 Reference Angle 33. Find the exact value of cos x AND the value of x in radians. Answer: You should recognize these coordinates from a 30-60-90 ∆ with hyp = 1 (In unit circle, cos is simply the x-coordinate.) cos x = and x = x ReferenceAngle must be 60o NOTE: This is not a special ∆ we have memorized, so to solve for x you would need a calculator to approximate cos-1(3/5). (Result is approx. 53.1o)

  18. Match each basic trig function with its graph. B. A. D. C. E. F. Answers: B E A D C F • y = cos x • y = sec x • y = sin x • y = csc x • y = tan x • y = cot x

  19. SUMMARY: (Use next slide to check your understanding) • Trig Function Transformations • y = ±a sin b(x ±c) ±d • |a|amplitude • a < 0 vertical reflection • 2/bperiod for sin, cos, csc & sec • /bperiodfor tan & cot • - chorizontal translation • (shifts in opposite direction of the ± sign ) • dvertical translation • (shifts in same direction of the ± sign )

  20. 35. What is the difference in the graphs of y = sin x and y = - sin x? Answer: Second graph is a vertical reflection of the first. 36. What is the difference in the graphs of y = 3 cos x and y = 2 cos x? Answer: amplitude (max & min displacement) of first is 3; its graph passes through the pt (0,3) . The amplitude of the second is 2; it passes through the pt (0, 2). 37. What is the difference in the graphs of y = sin x and y = sin x - 5? Answer: Second graph is shifted five unit down. First graph passes through the pt (0,0) and the second passes through (0, -5) *Note: this is very different from y = sin (x – 5) which is a horiz shift to the right. 38. What is the difference in the graphs of y = cos 4x and y = cos (x/2)? Answer: First has a shortened period of 2/4 = /2whereas the second has a lengthened period of(2)/(/2) = 4

  21. What is the difference in the graphs of • y = cos (x + )and y = cos (x - )? Answer: First graph is shifted  units to the left beginning at (-, 1) and the second graph is shifted  units right beginning at (, 1), so when you repeat cycles in both directions, the two graphs are exactly THE SAME – NO DIFFERENCE! • What is the difference in the graphs of • y = sin x and y = cos (x + /2)? Answer: when you shift the cos graph /2 units to the left, it lands on top of y = sin x. The two graphs are exactly THE SAME – NO DIFFERENCE!

  22. 41. What points do y = sin x and y = csc x share in common? Why? Answer: (-/2, -1), (/2, 1),(3/2, -1),(5/2, 1),…because the reciprocal of 1 stays at 1 and the reciprocal of -1 stays at -1. 42. Where are the vertical asymptotes for y = csc x? Why? Answer: x = -, , 3, 5,…because the reciprocal of 0 is undefined and the reciprocals of the very small fractional sine values close to these locations become infinitely large values that go towards ∞ and - ∞. 43. What hints might you give someone to graph y = -5 sec (2x + ) – 1? • Answer: • Make a dotted graph of y = -5 cos (2x + ) – 1 • Keep the max & min pts fixed • make vertical asymptotes through the (transformed locations of the) x-intercepts • Flip over the cos curves into U-type curves.

  23. 44. Where are the vertical asymptotes for y = tan x? Why? Answer: x = - /2, x = /2, x = 3/2, x = 5/2,… because tan = y/x or sin/cos and this is undefined whenever the x-coordinate is zero. 45. Where are the vertical asymptotes for y = cot x? Why? Answer: x = - , x = 0, x = , x = 2, x = 3,… because cot = x/y or cos/sin and this is undefined whenever the y-coordinate is zero. 46. Once you know where the asymptotes are, what is the other visual difference between the graphs of y = tan x and y = cot x? Answer: y = tan x goes up to the right and down to the left(since tangents are pos in quadrant 1 between 0 and /2 and neg in quad 4 between -/2 and0) y = cot x goes up to the left and down to the right(since cotangents are also pos in quadrant 1 between 0 and /2 and neg in quad 4 between -/2 and0)

  24. 47. Again, sketch a graph of each function. Click to see answers. y = sin x y = cos x Answers: y = tan x y = csc x y = sec x y = cot x • y = sin x • y = cos x • y = tan x • y = sin x • y = csc x • y = cot x

  25. 48. Graph y = -5 cos (x/2 – π/4) Answer: Factor b to rewrite as y = -5 cos ½ (x -π/2) Amp = 5 Vertical reflection Period: Phase start: π/2 Phase end: 9π/2 Fix your graph if needed, then click to see answer. 9 /2 5 /2  /2 3/2 7 /2

  26. 49. What function is graphed below? (There are many possible answers – you only need to find one.) Possible Answers Include:  /4  /2 Hints

  27. Answer: we restrict the domain to (from a minimum to a maximum sine value) where no x and no y will be repeated. • We’re close to the end! What is another symbolic way to write • y = arc sin x and how is your answer read aloud? Answer: y = sin-1x which is read “y is equal to inverse sine of x” 51. Do all functions have inverses? Answer: No, only functions that are one-to-one meaning each unique x is paired with a unique y. (No repeats on x or y, so the graph must pass both the vertical and the horizontal line tests.) 52. Since y = sin x has many repeated y values (imagine a horizontal line passing through all through those humps), how can there be an arc sin or inverse sine function?

  28. 53. Inverse functions are easy to graph if you recall that f -1(x) is a reflection of f(x) across what line? Answer: y = x or the 450 diagonal line 54. This means that the points (-π/2, -1) and (π/2, 1)on the graph of y = sin x, are reflected to (?, ?) and (?, ?) on y = arc sin x? Answer: (-1, - π/2) and (1, π/2) 55. Sketch the graph of y = sin -1 x Hint: sketch y = sin x first & reflect a few key pts across y = x. Click to see the answer (blue graph).

  29. 56. Visualize y = cos x. What restriction on the domain (close to the origin) will produce a one-to-one section with no repeats on x or y? Hint: always go from a minimum to a maximum height or max to min. Answer: 0 ≤ x ≤ π 57. This means that the points (0, 1) and (π, -1)on the graph of y = cos x, are reflected to (?, ?) and (?, ?) on y = arc cos x? Answer: (1, 0) and (-1, π) 58. Sketch the graph of y = cos -1 x Click to see the answer (blue graph).

  30. 59. Visualize y = tan x. What happens to a vertical asymptote, when it is reflected across y = x? Answer: (same as for arc sin) Answer: it becomes a horizontal asymptote. 60. What restriction on the domain of y = tan x would produce a one-to-one section? • Sketch the graph • of y = tan -1 x • Click to see • the answer • (blue graph).

  31. 62. To solve inverse trig functions, think in reverse order: what angle has that value as an answer. That is, n = arc sin ½ simply translates to “n is an angle that has a sin of ½” so we know n = π/6! Why do we know n cannot equal 5π/6? 63. Solve for y, given 64. Solve for r, given Answer: because the arc sin function is limited to the one-to-one interval from -π/2 to π/2. Answer: -π/4 (You cannot answer 5π/4 or 7π/4 because the arc sin function is limited to the one-to-one interval from -π/2 to π/2. ) Answer: 5π/6 *Always remember arc sin and arc tan functions are limited to the one-to-one interval from -π/2 to π/2 and arc cos is limited to 0 to π .

  32. 65. Now practice in random order. Visualize a unit circle, but don’t waste time drawing it. Practice until you can do these quickly and confidently! Click to check your answers.

  33. What geometry equation must be true about x & y in any right triangle like that shown below? Answer: x2 + y2 = 1 (x,y) 67. What “trig identity” does this equation become? hyp = 1 y= opp Answer: sin2a + cos2a= 1 which is called the Pythagorean Identity for obvious reason! a adj =x 68. What “trig identity” does this equation become if you divide through by cos2a? Answer: tan2a + 1= sec2a Also called an Pythagorean Identity 69. What “trig identity” would it have become if you divided through by sin2a? Answer: 1 + cot2a = csc2a Also called an Pythagorean Identity

  34. 70. If you reflect angle “a” vertically, what changes are made in the three trig ratios? -a b 1 y a x cos a Answer: cos (-a) = sin (-a) = tan (-a) = (x,y) - sin a - tan a 1 *Called “OPPOSITE ANGLE identities” 71. What geometry term applies to the pair of acute angles a & b in a right triangle (like the pink triangle below)? a Answer: “complementary” Sum of 90o but we now prefer to say b = (π/2 – a) radians (x,-y) 72. What “COFUNCTION trig identities” relate the ratios for COmplementary angles? Answer: sin a = cos(π/2 – a) csc a = sec(π/2 – a) cos a = sin(π/2 – a) sec a = csc(π/2 – a) tan a = cot(π/2 – a) cot a = tan(π/2 – a) bπ/2-a

  35. 73. Can you give the Double Angle Formulas? (Memory works, but as long as you recognize this and know where to find it quickly, you’re probably OK.This is even more true of others like half-angle, sum & difference, etc. which we will seldom use.) Answer: Answer: Answer: 74. Can you give the Law of Sines Formula? 75. Can you give the Law of Cosines Formulas?

  36. 76. Find the solutions of the equation in [0, 2π): Solution: Solution: • 77. Find ALL solutions of the equation: • sec2 t – 2 tan t = 0

  37. 78. What term is used to represent any function whose graph has repeated crests and troughs? Answer: sinusoidal 79. Use geometry to explain why all 26o angles have the same sine value? Answer: All right triangles with a 26o angle are similar to each other by AA theorem and we know the ratios of corresponding sides of similar polygons are equal . Thus, for all 26o angled rt triangles, the ratio of opp/hyp will be the same! Trigonometry is simply a study of ratios in similar triangles. 80. One more time, what are the definitions of the sin, cos and tan in any size right triangle and their definitions in the unit circle when the hypotenuse has a length of 1? Answer: Right triangles  SOHCAHTOA Unit circle cos = x; sin = y; tan = y/x

  38. THE END Congratulations! Think trig is hard? Think again! Check out this brief history of trig to see that folks have been doing trig for thousands of years! You can learn it too! • Repeat as needed until you remember your trig basics. • Look at the links on our class website to find more trig lessons and practice if needed. • You MUST know your trig, to be successful in calculus!

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