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we can break it up into the sum or difference of two of the special angles. See Table 1. Sum formula for cosine Difference formula for cosine cos(α + β) = cos α cos β − sin α sin β cos(α − β) = cos α cos β + sin α sin β Table 1 First, we will prove the difference formula for cosines. Let’s consider two points on the u... |
√ = √ ——— —— (cos2(α − β) + sin2(α − β)) − 2 cos(α − β) + 1 1 − 2 cos(α − β) + 1 2 − 2 cos(α − β) — Because the two distances are the same, we set them equal to each other and simplify. √ —— 2 − 2 cos α cos β − 2 sin α sin β = √ 2 − 2 cos α cos β − 2 sin α sin β = 2 − 2 cos(α − β) 2 − 2 cos(α − β) — Finally we subtrac... |
− . Find the exact value of cos 4 3 Example 2 Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine Find the exact value of cos(75°). Solution As 75° = 45° + 30°, we can evaluate cos(75°) as cos(45° + 30°). Thus, cos(45° + 30°) = cos(45°)cos(30°) − sin(45°)sin(30°) = = = — — — — 2 3 2 − ... |
2, sin(30°) = 1 √ __ ____ 2 2 Now we can substitute these values into the equation and simplify. 2 3 2 √ − 1 √ √ __ ____ ____ ____ 2 2 2 2 — 6 − √ 2 √ _________ 4 sin(45° − 30°) = = — — — — b. Again, we write the formula and substitute the given angles. sin(α − β) = sin α cos β − cos α sin β sin(135° − 120°) =... |
1 − sin2 β ______ 9 __ 1 − 25 ___ 16 __ 25 = √ = √ = 4 __ 5 Using the sum formula for sine, 3 1 __ __ = sin(α + β) sin cos−1 + sin−1 5 2 — = = sin α cos β + cos α sin β 3 4 √ 3 + 1 ____ __ __ __ · · 2 5 2 5 — 3 + 3 4 √ ________ 10 = Using the Sum and Difference Formulas for Tangent Finding exact values for the tan... |
an Expression Involving Tangent π π __ __ + . Find the exact value of tan 4 6 Solution Let’s first write the sum formula for tangent and substitute the given angles into the formula. tan(α + β) = tan α + tan β __ 1 − tan α tan β π π __ __ = + tan 4 6 π π __ __ + tan tan 4 6 ___ π π __ __ 1 − tan ... |
quadrant. See Figure 5. Again, using the Pythagorean Theorem, we have and π < β < 5 __ 13, the side adjacent to β is −5, the hypotenuse is 13, and β is in the third (−5)2 + a2 = 132 25 + a2 = 169 a2 = 144 a = ±12 Since β is in the third quadrant, a = –12. (–5, 0) y x β 13 (–5, –12) Figure 5 The next step is finding th... |
To find tan(α − β), we have the values we need. We can substitute them in and evaluate. tan(α − β) = tan α − tan β __ 1 + tan α tan β = = 12 3 __ __ − 5 4 __ 1 + 3 12 __ __ 5 4 − 33 __ 20 _ 56 __ 20 = − 33 _ 56 Analysis β are angles in the same triangle, which of course, they are not. Also note that A common mista... |
π __ − θ cot θ = tan 2 Notice that the formulas in the table may also be justified algebraically using the sum and difference formulas. For example, using we can write cos(α − β) = cos αcos β + sin αsin β, π π π __ __ __ cos θ + sin − θ = cos cos sin θ 2 2 2 = (0)cos θ + (1)sin θ = sin θ Example 7 Finding a Co... |
= sin α cos β + cos α sin β sin(α − β) = sin α cos β − cos α sin β We can rewrite each using the sum and difference formulas. sin(α + β) + sin(α − β) = sin α cos β + cos α sin β + sin α cos β − cos α sin β We see that the identity is verified. = 2 sin α cos β Example 9 Verifying an Identity Involving Tangent Verify th... |
wire S attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle α between the wires. See Figure 8. 47 ft 40 ft R S α β 50 ft Figure 8 Solution Let’s first summarize the information we can gather from the diagram. As only the sides adjacent to the righ... |
SeCTIOn exeRCISeS VeRBAl 1. Explain the basis for the cofunction identities and when they apply. 3. Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for f(x) = sin(x) and g(x) = cos(x). (Hint: 0 − x = −x) Al... |
2 4 3, and cos b = 1 21. Given that sin a = 4 π __ __ __ , find sin(a − b) and cos(a + b)., with a and b both in the interval 0, 2 3 5 For the following exercises, find the exact value of each expression. 1 __ 22. sin cos−1(0) − cos−1 2 1 1 __ __ − cos−1 24. tan sin−1 2 2 3 2 + sin− 1 23... |
x 37. f(x) = sin(2x), g(x) = 2 sin x cos x 38. f(θ) = cos(2θ), g(θ) = cos2 θ − sin2 θ 39. f(θ) = tan(2θ), g(θ) = tan θ _ 1 + tan2θ 41. f(x) = tan(−x), g(x) = tan x − tan(2x) __ 1 − tan x tan(2x) TeCHnOlOGY 40. f(x) = sin(3x)sin x, g(x) = sin2(2x)cos2 x − cos2(2x)sin2 x For the following exercises, find the exact value... |
the same triangle, then prove or disprove: tan α + tan β + tan γ = tan α tan β tan γ. 584 CHAPTER 7 trigonometric identities and eQuations leARnInG OBjeCTIVeS In this section, you will: • • • • Use double-angle formulas to find exact values. Use double-angle formulas to verify identities. Use reduction formulas to sim... |
cos(2θ) = cos2 θ − sin2 θ = (1 − sin2 θ) − sin2 θ = 1 − 2sin2 θ cos(2θ) = cos2 θ − sin2 θ = cos2 θ − (1 − cos2 θ) = 2 cos2 θ − 1 SECTION 7.3 douBle-angle, halF-angle, and reduction Formulas 585 Similarly, to derive the double-angle formula for tangent, replacing α = β = θ in the sum formula gives tan(α + β) = tan α + ... |
(−4)2 + (3)2 = c2 16 + 9 = c2 25 = c2 c = 5 Now we can draw a triangle similar to the one shown in Figure 2. y (–4, 3) (–4, 0) 5 θ Figure 2 x 586 CHAPTER 7 trigonometric identities and eQuations a. Let’s begin by writing the double-angle formula for sine. sin(2θ) = 2 sin θ cos θ We see that we to need to find sin θ an... |
− 1 Analysis This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function. Using Double-Angle Formulas to Verify Identities Establishing i... |
tan θ − Analysis Here is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show = 2 _____... |
cos(2θ) = 2 cos2 θ − 1. Solve for cos2 θ: sin2 θ = cos(2θ) = 2 cos2 θ − 1 1 + cos(2θ) = 2 cos2 θ 1 + cos(2θ) _________ 2 = cos2 θ The last reduction formula is derived by writing tangent in terms of sine and cosine: tan2 θ = sin2 θ _____ cos2 θ 1 − cos(2θ) _________ 2 __ 1 + cos(2θ) _________ 2 1 − cos(2θ) _______... |
the power-reducing formulas to prove sin3 (2x) = 1 __ sin(2x) [1 − cos(4x)] 2 Solution We will work on simplifying the left side of the equation: 1 − cos(4x) _________ 2 sin3(2x) = [sin(2x)][sin2(2x)] = sin(2x) 1 __ [1 − cos(4x)] = sin(2x) 2 1 __ = [sin(2x)][1 − cos(4x)] 2 Substitute the power-reduction fo... |
2 α __ 1 + cos 2 ⋅ 2 2 __ = 1 + cos α ________ 2 _________ 1 + cos α ________ 2 α __ cos 2 = ± √ 590 CHAPTER 7 trigonometric identities and eQuations For the tangent identity, we have tan2 θ = 1 − cos(2θ) _________ 1 + cos(2θ) α __ = tan2 2 = = ± √ α __ tan 2 α __ 1 − cos 2 ⋅ 2 __ α __ 1 + cos 2... |
SECTION 7.3 douBle-angle, halF-angle, and reduction Formulas 591 Example 8 Finding Exact Values Using Half-Angle Identities Given that tan α = and α lies in quadrant III, find the exact value of the following: 8 __ 15 α __ b. cos 2 α __ a. sin 2 α __ c. tan 2 Solution Using the given information, we can dr... |
· 17 2 ___ 1 __ = ± √ 17 — 17 √ _ 17 = − α __ We choose the negative value of cos because the angle is in quadrant II because cosine is negative in 2 quadrant II. c. α __ To find tan, we write the half-angle formula for tangent. Again, we substitute the value of the cosine we 2 found from the triangle in Figure 3 and ... |
_ tan 2 —————— — 34 3 √ _ 1 − 34 __ 34 3 √ _ 1 + 34 — —————— — 34 34 − 3 √ __ 34 __ 34 + 3 √ __ 34 34 — ___________ — 34 − 3 √ 34 __________ — 34 + 3 √ 34 = √ = √ = √ ≈ 0.57 We can take the inverse tangent to find the angle: tan−1 (0.57) ≈ 29.7°. So the angle of the ramp for novice competition is ≈ 29.7°. Access these... |
7. If cos x = − 1 __, and x is in quadrant III. 2 8. If tan x = −8, and x is in quadrant IV. For the following exercises, find the values of the six trigonometric functions if the conditions provided hold. 9. cos(2θ) = 3 __ and 90° ≤ θ ≤ 180° 5 and 180° ≤ θ ≤ 270° 10. cos(2θ) = 1 _ — 2 √ For the following exercises, s... |
cos(2α), and tan(2α). α __ α __ α __ , and tan , cos 27. Find sin . 2 2 2 For the following exercises, simplify each expression. Do not evaluate. 28. cos2(28°) − sin2(28°) 31. cos2(9x) − sin2(9x) 29. 2 cos2(37°) − 1 32. 4 sin(8x) cos(8x) 30. 1 − 2 sin2(17°) 33. 6 sin(5x) cos(5x) For the following exercises, p... |
= 1 − tan2 α ________ 1 + tan2 α 58. (sin2 x − 1)2 = cos(2x) + sin4 x 59. sin(3x) = 3 sin x cos2 x − sin3 x 60. cos(3x) = cos3 x − 3 sin2 x cos x 61. 1 + cos(2t) ___________ = sin(2t) − cos t 2 cos t ________ 2 sin t − 1 62. sin(16x) = 16 sin x cos x cos(2x)cos(4x)cos(8x) 63. cos(16x) = (cos2 (4x) − sin2 (4x) − sin(8x... |
) + cos(α + β) Then, we divide by 2 to isolate the product of cosines: 1 __ [cos(α − β) + cos(α + β)] cos α cos β = 2 SECTION 7.4 sum-to-product and product-to-sum Formulas 597 How To… Given a product of cosines, express as a sum. 1. Write the formula for the product of cosines. 2. Substitute the given angles into the ... |
the following product as a sum containing only sine or cosine and no products: sin(4θ)cos(2θ). Solution Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify. 1 __ sin α cos β = [sin(α + β) + sin(α − β)] 2 1 __ [sin(4θ + 2θ) + sin(4θ − 2θ)] sin(4θ)cos(2θ) ... |
[cos(α − β) + cos(α + β)] 2 1 __ [cos(3θ − 5θ) + cos(3θ + 5θ)] cos(3θ)cos(5θ) = 2 1 __ [cos(2θ) + cos(8θ)] = 2 Use even-odd identity. Try It #3 Use the product-to-sum formula to evaluate cos 11π ___ 12 cos π __. 12 expressing Sums as Products Some problems require the reverse of the process we just used. The sum-to-pr... |
� α + β _____ 2 cos α + cos β = 2 cos α + β _____ 2 cos α − β _____ 2 Example 4 Writing the Difference of Sines as a Product Write the following difference of sines expression as a product: sin(4θ) − sin(2θ). Solution We begin by writing the formula for the difference of sines. sin α − sin β = 2 sin α − β _... |
− cos(2t) _____________ sin(4t) + sin(2t) = −2 sin ____ cos 2 sin 4t + 2t ______ 2 4t + 2t ______ 2 4t − 2t ______ 2 4t − 2t ______ 2 sin = −2 sin(3t)sin t ___________ 2 sin(3t)cos t = −2 sin(3t)sin t ___________ 2 sin(3t)cos t = − sin t ____ cos t = −tan t Analysis Recall that verifying trigonometric ... |
we would convert an equation from a sum to a product and give an example. 2. Explain two different methods of calculating cos(195°)cos(105°), one of which uses the product to sum. Which method is easier? 4. Explain a situation where we would convert an equation from a product to a sum, and give an example. AlGeBRAIC F... |
31. sin(−1°) + sin(−2°) For the following exercises, prove the identity. 32. cos(a + b) ________ = cos(a − b) 1 − tan a tan b ___________ 1 + tan a tan b 34. 6cos(8x)sin(2x) ____________ sin(−6x) = −3 sin(10x)csc(6x) + 3 33. 4sin(3x)cos(4x) = 2 sin(7x) − 2 sinx 35. sin x + sin(3x) = 4sin x cos2 x 36. 2(cos3 x − cos x ... |
x 48. sin(2x) + sin(4x) _____________ sin(2x) − sin(4x) = −tan(3x)cot x For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. 49. sin(9t) − sin(3t) _____________ cos(9t) + cos(3t) 51. sin(3x) − sin x ____________ sin... |
G OBjeCTIVeS In this section, you will: • • • • • • • Solve linear trigonometric equations in sine and cosine. Solve equations involving a single trigonometric function. Solve trigonometric equations using a calculator. Solve trigonometric equations that are quadratic in form. Solve trigonometric equations using fundam... |
for stating all possible solutions for a function where the period is 2π: sin θ = sin(θ ± 2kπ) There are similar rules for indicating all possible solutions for the other trigonometric functions. Solving trigonometric equations requires the same techniques as solving algebraic equations. We read the equation from left... |
θ < 2π. Solution Use algebraic techniques to solve the equation. 2cos θ − 3 = −5 2cos θ = −2 cos θ = −1 θ = π Try It #1 Solve exactly the following linear equation on the interval [0, 2π): 2sin x + 1 = 0. Solving equations Involving a Single Trigonometric Function When we are given equations that involve only one of t... |
1 ____ sin θ sin θ = − 1 __ 2 θ = 7π ___, 6 23π 19π 11π ___ ___ ___,, 6 6 6 Analysis As sin θ = − 1 _, notice that all four solutions are in the third and fourth quadrants. 2 Example 6 Solving an Equation Involving Tangent π __ = 1, 0 ≤ θ < 2π. Solve the equation exactly: tan θ − 2 π __ Solution Recall that the ta... |
θ = 0.8, where θ is in radians. Solution Make sure mode is set to radians. To find θ, use the inverse sine function. On most calculators, you will need to push the 2ND button and then the SIN button to bring up the sin−1 function. What is shown on the screen is sin−1(. The calculator is ready for the input within the ... |
In quadrant III, the reference angle is θ´≈ π − 1.8235 ≈ 1.3181. The other solution in quadrant III is θ´ ≈ π + 1.3181 ≈ 4.4597. The solutions are θ ≈ 1.8235 ± 2πk and θ ≈ 4.4597 ± 2πk. Try It #3 Solve cos θ = −0.2. Solving Trigonometric equations in Quadratic Form Solving a quadratic equation may be more complicated,... |
2 θ = cos−1 — 13 −3 + √ _________ 2 — 13 −3 + √ _________ 2 θ = cos−1 ≈ 1.26 608 CHAPTER 7 trigonometric identities and eQuations This terminal side of the angle lies in quadrant I. Since cosine is also positive in quadrant IV, the second solution is θ = 2π − cos−1 ≈ 5.02 — 13 −3 + √ _________ 2 Try It #4 ... |
2x2 + x = 0 x(2x + 1) = 0 x = 0 (2x + 1) = 0 1 __ x = − 2 Then, substitute back into the equation the original expression sin θ for x. Thus, sin θ = 0 The solutions within the domain 0 ≤ θ < 2π are θ = 0, π, θ = 0, π sin θ = − 1 __ 2 11π 7π ___ ___, 6 6 11π ___. 6 7π ___, 6 θ = If we prefer not to substitute, we can s... |
6 sin θ = 1 π __ θ = 2 Try It #6 Solve the quadratic equation 2cos2 θ + cos θ = 0. Solving Trigonometric equations Using Fundamental Identities While algebra can be used to solve a number of trigonometric equations, we can also use the fundamental identities because they make solving equations simpler. Remember that t... |
__ ± 2πk; if cos θ = 1, then θ = 0 ± 2πk. 3 Example 16 Solving an Equation Using an Identity Solve the equation exactly using an identity: 3cos θ + 3 = 2sin2 θ, 0 ≤ θ < 2π. Solution If we rewrite the right side, we can write the equation in terms of cosine: 3cos θ + 3 = 2sin2 θ 3cos θ + 3 = 2(1 − cos2 θ) 3cos θ + 3 = ... |
, 2x = or 2x = and θ = Therefore, the possible angles are __ __ __ = cos Does this make sense? Yes, because cos 2 . 2 3 6 5π __ 3 π __, which means that x = or x = 6 5π __. 6 Are there any other possible answers? Let us return to our first step. π π __ __ In quadrant I, 2x =, so x = as noted. Let us revolve ar... |
the problem. One of the cables that anchors the center of the London Eye Ferris wheel to the ground must be replaced. The center of the Ferris wheel is 69.5 meters above the ground, and the second anchor on the ground is 23 meters from the base of the Ferris wheel. Approximately how long is the cable, and what is the ... |
a)2 b2 = (4a)2 − a2 b2 = 16a2 − a2 b2 = 15a2 Thus, the ladder touches the wall at a √ 15 15 feet from the ground. b = a √ — — Access these online resources for additional instruction and practice with solving trigonometric equations. • Solving Trigonometric equations I (http://openstaxcollege.org/l/solvetrigeqI) • Solv... |
� 5 — 3 14. 2cos θ = −1 17. 2sin(3θ) = 1 3 20. cos(2θ) = − √ ____ 2 — 6. 2cos θ = 1 9. tan x = 1 12. csc2 x − 4 = 0 15. 2sin θ = −1 18. 2sin(2θ) = √ — 3 21. 2sin(πθ) = 1 For the following exercises, find all exact solutions on [0, 2π). 23. sec(x)sin(x) − 2sin(x) = 0 24. tan(x) − 2sin(x)tan(x) = 0 25. 2cos2 t + cos(t) =... |
1 = 0 45. 3cos2 x − 2cos x − 2 = 0 46. 5sin2 x + 2sin x − 1 = 0 47. tan2 x + 5tan x − 1 = 0 48. cot2 x = −cot x 49. −tan2 x − tan x − 2 = 0 SECTION 7.5 section exercises 615 For the following exercises, find exact solutions on the interval [0, 2π). Look for opportunities to use trigonometric identities. 50. sin2 x − c... |
values on the interval [0, 2π). Round to four decimal places. 77. tan2 x + 3tan x − 3 = 0 78. 6tan2 x + 13tan x = −6 79. tan2 x − sec x = 1 80. sin2 x − 2cos2 x = 0 81. 2tan2 x + 9tan x − 6 = 0 82. 4sin2 x + sin(2x)sec x − 3 = 0 exTenSIOnS For the following exercises, find all solutions exactly to the equations on the... |
feet long. What is the angle of elevation of the sun? 103. A spotlight on the ground 3 feet from a 5-foot tall woman casts a 15-foot tall shadow on a wall 6 feet from the woman. At what angle is the light? 94. If a loading ramp is placed next to a truck, at a height of 4 feet, and the ramp is 15 feet long, what angle ... |
New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function. Many other natural phenomena are... |
maximum or minimum value of the function. 618 CHAPTER 7 trigonometric identities and eQuations Showing How the Properties of a Trigonometric Function Can Transform a Graph Example 1 π _ Show the transformation of the graph of y = sin x into the graph of y = 2sin 4x − + 2. 2 Solution Consider the series of graphs i... |
4 π _ b. y = −3sin 2x + 2 c. y = cos x + 3 Solution We will solve these problems according to the models. 1 _ x involves sine, so we use the form a. y = 2sin 4 y = Asin(Bt − C) + D We know that ∣ A ∣ is the amplitude, so the amplitude is 2. Period is, so the period is 2π _ B 2π _ B = 2π _ 1 __ 4 = 8π See the... |
also indicate x-intercepts. For example, suppose we want to graph the function y = cos θ. We know that the period is 2π, so we find the interval between key points as follows. 2π _ 4 π _ = 2 π _ Starting with θ = 0, we calculate the first y-value, add the length of the interval to 0, and calculate the second 2 π _ rep... |
data (round to the nearest tenth) and sketch the graph. SECTION 7.6 modeling with trigonometric eQuations 621 Month January February March April May June July August September October November December Temperature, ° F 42.5 44.5 48.5 52.5 58 63 68.5 69 64.5 55.5 46.5 43.5 Solution Recall that amplitude is found using ... |
of the hour hand to the ceiling x hours after noon. Find the equation that models the motion of the clock and sketch the graph. Solution Begin by making a table of values as shown in Table 4. x Noon 3 PM 6 PM 9 PM Midnight y 30 in 54 in 78 in 54 in 30 in Table 4 Points to plot (0, 30) (3, 54) (6, 78) (9, 54) (12, 30) ... |
t + 11 6 (0, 15) (12, 15) Midline: y = 11 (6, 7) 16 14 12 10 10 11 12 Time Figure 10 Try It #3 The daily temperature in the month of March in a certain city varies from a low of 24°F to a high of 40°F. Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperat... |
hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of ... |
is. 4 4. See Figure 14. Damped Harmonic Motion y 5 3 1 –1 –3 –5 π t y = 5cos 2 1 2 3 4 t Figure 14 In reality, a pendulum does not swing back and forth forever, nor does an object on a spring bounce up and down forever. Eventually, the pendulum stops swinging and the object stops bouncing and both return to equilibriu... |
7 trigonometric identities and eQuations The second spring system has a damping factor of c = 0.1 and can be modeled as f (t) = 10e−0.1tcos(πt) Figure 16 models the motion of the second spring system. f(t) f(t) = 10e–0.1tcos (πt) 12 8 4 –12 –8 –4 4 8 12 t –4 –8 –12 Figure 16 Analysis Notice the differing effects of th... |
a. As period is 2π _ ω, we have π 2π _ _ = ω 6 ωπ = 6(2π) ω = 12 The damping factor is given as 10 and the amplitude is 7. Thus, the model is y = 7e−10tsin(12t). See Figure 19. y 3 2 1 –1 y = 7e–10t sin(12t) 1 2 3 4 t Figure 19 20 = ω _ 2π 40π = ω b. As frequency is, we have ω _ 2π The damping factor is given as 0.2 a... |
will write A as a negative value. We can write the amplitude portion of the function as We put (1 − 0.30)t in the form ect as follows: A(t) = 5(1 − 0.30)t 0.7 = ec c = ln 0.7 c = −0.357 Now let’s address the period. The spring cycles through its positions every 3 seconds, this is the period, and we can use the formula... |
als. ae−c(t + 3) = 1 _ ae−ct 2 e−ct · e−3c = 1 _ e−ct 2 e−3c = 1 _ 2 e3c = 2 e3c = 2 3c = ln(2) c = ln(2) _ 3 Then use the laws of logarithms. The damping constant is ln(2) ___. 3 Bounding Curves in Harmonic Motion Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude, such ... |
ations 7.6 SeCTIOn exeRCISeS VeRBAl 1. Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary? 3. If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not? 2. What information is nece... |
�s average yearly rainfall is currently 20 inches and varies seasonally by 5 inches. Due to unforeseen circumstances, rainfall appears to be decreasing by 15% each year. How many years from now would we expect rainfall to initially reach 0 inches? Note, the model is invalid once it predicts negative rainfall, so choose... |
fluctuates between about 6 million square kilometers on September 1 to 14 million square kilometers on March 1. Assuming a sinusoidal fluctuation, when are there less than 9 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day. 24. The sea ice area around the South Pole fluctu... |
modeled by the function π _ h(t) = 4cos t , where t is measured in seconds. Find the amplitude, period, and frequency of this displacement. 2 For the following exercises, construct an equation that models the described behavior. 32. The displacement h(t), in centimeters, of a mass suspended by a spring is modeled b... |
pulled 19 cm down from equilibrium and released. After 3 seconds, the amplitude has decreased to 13 cm. The spring oscillates 14 times each second. Find a function that models the distance, D, the end of the spring is from equilibrium in terms of seconds, t, since the spring was released. from equilibrium and released... |
time. The first spring, which oscillates 14 times per second, was initially pulled down 2 cm from equilibrium, and the amplitude decreases by 8% each second. The second spring, oscillating 22 times per second, was initially pulled down 10 cm from equilibrium and after 3 seconds has an amplitude of 2 cm. Which spring c... |
trigonometric identity that allows the writing of a product of trigonometric functions as a sum or difference of trigonometric functions Pythagorean identities set of equations involving trigonometric functions based on the right triangle properties quotient identities pair of identities based on the fact that tangent... |
− tan β ___________ 1 + tan α tan β π _ sin θ = cos − θ 2 π _ − θ cos θ = sin 2 π _ − θ tan θ = cot 2 π _ − θ cot θ = tan 2 π _ − θ sec θ = csc 2 π _ − θ csc θ = sec 2 sin(2θ) = 2sin θ cos θ cos(2θ) = cos2 θ − sin2 θ = 1 − 2sin2 θ = 2cos2 θ − 1 tan(2θ) = 2tan θ ________ 1 − tan2 θ sin2θ = 1 − c... |
2 α + β _____ 2 α + β _____ 2 cos α − β _____ 2 α + β _____ 2 α − β _____ 2 α − β _____ 2 sin Standard form of sinusoidal equation y = A sin(Bt − C) + D or y = A cos(Bt − C) + D Simple harmonic motion Damped harmonic motion Key Concepts d = a cos(ωt) or d = a sin(ωt) f (t) = ae−ct sin(ωt) or f (t) = ae... |
and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of th... |
. See Example 1, Example 2, and Example 3. • We can also derive the sum-to-product identities from the product-to-sum identities using substitution. • We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See Example 4. 638 C... |
2. • Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See Example 3. • Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See Example 4, Example 5,... |
cos(23°) + sin(83°)sin(23°) For the following exercises, prove the identity. 15. cos(4x) − cos(3x)cosx = sin2 x − 4cos2 x sin2 x 16. cos(3x) − cos3 x = − cos x sin2 x − sin x sin(2x) For the following exercise, simplify the expression. 1 1 __ __ x + tan x tan 8 2 ___ 1 1 __ __ x x tan 1 − tan 2 8 17. Fo... |
. 2cos(2x) _______ sin(2x) = cot x − tan x 27. cot x cos(2x) = − sin(2x) + cot x For the following exercises, rewrite the expression with no powers. 28. cos2 x sin4(2x) 29. tan2 x sin3 x SUM-TO-PRODUCT AnD PRODUCT-TO-SUM FORMUlAS For the following exercises, evaluate the product for the given expression using a sum or ... |
[0, 2π). Round to four decimal places. 46. √ — 3 cot2 x + cot x = 1 47. csc2 x − 3csc x − 4 = 0 For the following exercises, graph each side of the equation to find the zeroes on the interval [0, 2π). 48. 20cos2 x + 21cos x + 1 = 0 49. sec2 x − 2sec x = 15 MODelInG WITH TRIGOnOMeTRIC eQUATIOnS For the following exerci... |
. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by 5% each year. Find a function modeling the population of native fish with respect to t, the number of years from now. Also determine how many years it will take for the carp to overtake the native fish popul... |
. Rewrite the expression sin4 x with no powers greater than 1. 17. tan3 x − tan x sec2 x = tan(−x) 18. sin(3x) − cos x sin(2x) = cos2 x sin x − sin3 x 19. sin(2x) ______ − sin x cos(2x) ______ cos x = sec x 20. Plot the points and find a function of the form y = Acos(Bx + C) + D that fits the given data. x y 0 −2 1 2 2... |
the docks are 2 feet above current water levels, at what point will the water first rise above the docks? 8 Further Applications of Trigonometry Figure 1 General Sherman, the world’s largest living tree. (credit: Mike Baird, Flickr) CHAPTeR OUTlIne 8.1 non-right Triangles: law of Sines 8.2 non-right Triangles: law of ... |
° 20 miles Figure 1 Using the law of Sines to Solve Oblique Triangles In any triangle, we can draw an altitude, a perpendicular line from one vertex to the opposite side, forming two right triangles. It would be preferable, however, to have methods that we can apply directly to non-right triangles without first having ... |
Law of Sines. sin α _ a = sin β _ b = sin γ _ c Note the standard way of labeling triangles: angle α (alpha) is opposite side a; angle β (beta) is opposite side b; and angle γ (gamma) is opposite side c. See Figure 6. While calculating angles and sides, be sure to carry the exact values through to the final answer. Ge... |
set of angles and sides is α = 50° β = 100° γ = 30° a = 10 b ≈ 12.9 c ≈ 6.5 Try It #1 Solve the triangle shown in Figure 8 to the nearest tenth. β c a 43° 98° α 22 Figure 8 Using The law of Sines to Solve SSA Triangles We can use the Law of Sines to solve any oblique triangle, but some solutions may not be straightfor... |
49.9°, which means that β = 180° − 49.9° = 130.1°. (Remember that the sine function is positive in both the first and second quadrants.) Solving for γ, we have γ = 180° − 35° − 130.1° ≈ 14.9° 648 CHAPTER 8 Further applications oF trigonometry We can then use these measurements to solve the other triangle. Since γ' is ... |
9 9sin(85°) _ 12 = sin β Isolate the unknown. SECTION 8.1 non-right triangles: law oF sines 649 To find β, apply the inverse sine function. The inverse sine will produce a single result, but keep in mind that there may be two values for β. It is important to verify the result, as there may be two viable solutions, onl... |
915 sin α = α 4 50° 10 Figure 14 We can stop here without finding the value of α. Because the range of the sine function is [−1, 1], it is impossible for the sine value to be 1.915. In fact, inputting sin−1 (1.915) in a graphing calculator generates an ERROR DOMAIN. Therefore, no triangles can be drawn with the provide... |
90)(52)sin(102°) 2 Area ≈ 2289 square units Try It #5 Find the area of the triangle given β = 42°, a = 7.2 ft, c = 3.4 ft. Round the area to the nearest tenth. Solving Applied Problems Using the law of Sines The more we study trigonometric applications, the more we discover that the applications are countless. Some are... |
onometric applications. • law of Sines: The Basics (http://openstaxcollege.org/l/sinesbasic) • law of Sines: The Ambiguous Case (http://openstaxcollege.org/l/sinesambiguous) 652 CHAPTER 8 Further applications oF trigonometry 8.1 SeCTIOn exeRCISeS VeRBAl 1. Describe the altitude of a triangle. 3. When can you use the La... |
8.2, a = 11.3 15. γ = 113°, b = 10, c = 32 18. a = 20.5, b = 35.0, β = 25° 21. b = 13, c = 5, γ = 10° 16. b = 3.5, c = 5.3, γ = 80° 19. a = 7, c = 9, α = 43° 22. a = 2.3, c = 1.8, γ = 28° For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles i... |
° 15 4.5 51° 2.9 47. 58° 9 11 51° 48. 49. 25 18 40° 30° 3.5 115° 30 50 654 CHAPTER 8 Further applications oF trigonometry exTenSIOnS 50. Find the radius of the circle in Figure 18. Round to 51. Find the diameter of the circle in Figure 19. Round the nearest tenth. to the nearest tenth. 145° 3 8.3 110° Figure 18 Figure ... |
shows a satellite orbiting Earth. The satellite passes directly over two tracking stations A and B, which are 69 miles apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 86.2° and 83.9°, respectively. How far is the satellite from station A and how high ... |
300 feet closer to the building and find the angle of elevation to be 50°. Assuming that the street is level, estimate the height of the building to the nearest foot. 67. In order to estimate the height of a building, two students stand at a certain distance from the building at street level. From this point, they fin... |
being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road, the park measures 215 feet. 72. Three cities, A, B, and C, are located so that city A is due east of city B. If city C is located 35° west of north from city B and is 100 miles fr... |
-side) triangle. In this section, we will investigate another tool for solving oblique triangles described by these last two cases. Using the law of Cosines to Solve Oblique Triangles The tool we need to solve the problem of the boat’s distance from the port is the Law of Cosines, which defines the relationship among a... |
2) + b2 sin2 θ Expand the perfect square. = b2 cos2 θ + b2 sin2 θ + c2 − 2bccos θ Group terms noting that cos2 θ + sin2 θ = 1. = b2(cos2 θ + sin2 θ) + c2 − 2bccos θ Factor out b2. a2 = b2 + c2 − 2bccos θ The formula derived is one of the three equations of the Law of Cosines. The other equations are found in a similar ... |
to find the measure of a second angle. 4. Compute the measure of the remaining angle. 660 CHAPTER 8 Further applications oF trigonometry Example 1 Finding the Unknown Side and Angles of a SAS Triangle Find the unknown side and angles of the triangle in Figure 4. γ b a 5 10 α c 5 12 Figure 4 30° β Solution First, make ... |
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