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mathematics
precalculus
Precalculus Concepts Through Functions A Unit Circle Approach To Trigonometry 5th Edition Michael Sullivan - Solutions
In Problems 27 – 38 , two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).a = 7, b = 14, A = 30°
In Problems 31–36, use the results of Problem 29 or 30 to find the area of each triangle. Round answers to two decimal places.A = 40°, B = 20°, a = 2
In Problems 29–42, use the right triangle shown below. Then, using the given information, solve the triangle.a = 6, A = 40°; find b, c, and B В C a A b
In Problems 27–34, graph each function by adding y-coordinates.f (x) = sin(2x) + cos x
In Problems 31–36, use the results of Problem 29 or 30 to find the area of each triangle. Round answers to two decimal places.A = 50°, C = 20°, a = 3
A highway whose primary directions are north–south is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The figure shows the path that they decide on and
In Problems 27–34, graph each function by adding y-coordinates.g(x) = sin x + sin(2x)
In Problems 27–34, graph each function by adding y-coordinates.g(x) = cos(2x) + cos x
In Problems 27–38, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).a = 8, c = 3, C = 125°
In Problems 31–36, use the results of Problem 29 or 30 to find the area of each triangle. Round answers to two decimal places.A = 110°, C = 30°, c = 3
The function y = −3 cos(6x) has amplitude______ and period _____.
Solve: x + 2 x - 3 < 2
In Problems 83–94, find the exact value of each expression. sec (2 tan-¹2)
In Problems 83–94, find the exact value of each expression. $²(/sin-1 cos 2 315 5/
(a) The area A of a regular octagon is given by the formulawhere r is the apothem, which is a line segment from the center of the octagon perpendicular to a side. See the figure. Find the exact area of a regular octagon whose apothem is 12 inches.(b) The area A of a regular octagon is also given by
In Problems 83–94, find the exact value of each expression. csc[2 sin-¹(-3)]
Refer to Problem 105. The figure shows that the interior angle of a regular dodecagon has measure 150°, and the apothem equals the radius of the inscribed circle.(a) Find the exact area of a regular dodecagon with side a = 5 cm.(b) Find the radius of the inscribed circle for the regular dodecagon
An oscilloscope often displays a sawtooth curve. This curve can be approximated by sinusoidal curves of varying periods and amplitudes. A first approximation to the sawtooth curve is given byShow that y = sin(2πx)cos2 (πx). y = sin(2x) + sin(4x)
Find the value of the number C: 1 2 1 sin²x + C = -cos (2x) 4
Find the value of the number C: 1 2 cos²x + C = cos(2x) 4
Graphby using transformations. f(x) = sin ² x 1 - cos (2x) for 0 ≤ x ≤ 2π 2
In Problems 95–100, find the real zeros of each trigonometric function on the interval 0 ≤ θ< 2π. f (x) = 2 sin2 x − sin(2x)f (x) = 2 sin2x − sin(2x)
f (x) = 3 sin x(a) Find the zeros of f on the interval [-2π, 4π]. (b) Graph f(x) = 3 sin x on the interval [-2π, 4π]. (c) Solve f(x) = 3/2 on the interval [-2π, 4π]. What points are on the graph of f? Label these points on the graph drawn in part (b). (d) Use the graph drawn in
If z = α/2, show that cosa = 1-z² 2 1+z²
If z = tan α/2, show that sin a 2z 1+z²
Repeat Problem 115 for g(x) = cos2 x.Data from problem 115Graphby using transformations. f(x) = sin ² x 1 - cos (2x) for 0 ≤ x ≤ 2π 2
In Problems 95–100, find the real zeros of each trigonometric function on the interval 0 ≤ θ< 2π.f (x) = sin(2x) + cos x
f (x) = 2 cos x(a) Find the zeros of f on the interval [-2π, 4π]. (b) Graph f(x) = 2 cosx on the interval [-2π, 4π]. (c) Solve f(x) = -√3 on the interval [-2π, 4π]. What points are on the graph of f? Label these points on the graph drawn in part (b). (d) Use the graph drawn
If cos(2x) + (2m − 1)sin x + m − 1= 0, find m so that there is exactly one real solution for 5x5 KIN
In Problems 95–100, find the real zeros of each trigonometric function on the interval 0 ≤ θ< 2π.f (x) = cos(2x) − 5 cos x − 2
Show that sin ³0+ sin ³ (0 + 120°) + sin³ (0 + 240°) 3 --sin (30) 4
Mixed Practice f (x) = cot x(a) Solve f(x) = -√3. (b) For what values of x is f(x) > -√3 on the interval (0, π)?
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.The functionis one-to-one. Find f−1. f(x) = 3 - x 2x 5
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Find the exact value of sin 2π sin- 3 4π cos- 3 ليا
In Problems 1 and 2, find the exact value of the six trigonometric functions of the angle θ in each figure. 0 3 4
In Problems 1 and 2, find the exact value of the six trigonometric functions of the angle θ in each figure. 4 18 2
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Find an equation of the line that contains the point (2, −3) and is perpendicular to the line y =
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Graph f (x) = −x2 + 6x + 7. Label the vertex and any intercepts.
Find the exact value of the six trigonometric functions of the angle θ in the figure. 3 6 0
True or False 0 cos²2 = 1+ sin 0 2
Find the domain of the function f(x) = √√x²-3x - 4
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Graph y = −2 cos(π/2 x). Show at least two periods.
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Find a polynomial function of degree 3 whose real zeros are −5, −2, and 2. Use 1 for the leading
In Problems 3–5, find the exact value of each expression. Do not use a calculator. sec 55° csc 35°
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Solve: 2x+7 = 3x+2
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Find the distance between the vertices of the parabolas f (x) = x2 − 4x − 1 and g(x) = −x2 −
Problems 123–132. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Find the average rate of change of f (x) = log2 x from 4 to 16.
In Problems 3–5, find the exact value of each expression. Do not use a calculator.cos62° − sin28°
Problems 123–132 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Solve for D: 6x − 5xD − 5y + 4yD + 3 − 4D = 0
Solve sin A= 1/2 if 0 ≤ A ≤ π.
Find the exact value of sin40° − cos50°.
If one side and two angles of a triangle are known, which law can be used to solve the triangle?(a) Law of Sines(b) Law of Cosines(c) Either a or b(d) The triangle cannot be solved.
Graph the function: y = 3 sin(πx)
The sum of the measures of the two acute angles in a right triangle is ______.(a) 45°(b) 90°(c) 180°(d) 360°
In Problems 6 and 7, solve each triangle. 10 20⁰ a A b
In Problems 6 and 7, solve each triangle. 5 B a A 2
If tanθ= −2 and 3π/2 < θ < 2π, find the exact value of: (a) sin 0 (d) cos(20) (b) cos (e) sin( sin (120) (c) sin(20) (1) cos(10)
Graph each the following functions: (a) y = x (d) y = x³ (g) y = sin x (b) y = x² (e) y = ex (h) y = cos x (c) y = √x (f) y = ln x (i) y = tan x
Given two sides of a triangle, b and c , and the included angle A , the altitude h from angle B to side b is given by _____. (a) ab sin A (b) b sin A (c) c sin A (d) (d)—be sin A 2
If two sides and the included angle of a triangle are known, which law can be used to solve the triangle?(a) Law of Sines(b) Law of Cosines(c) Either a or b(d) The triangle cannot be solved.
In Problems 59–63, use the information given about the angles α and β to find the exact value of: (a) sin(a + B) (e) sin(2a) (b) cos(a + 3) (f) cos(2/3) (c) sin(a - 3) В (g) sin sin 2 (d) tan(a + B) (h) cos 2
Graph the function: y = −2 cos(2x −π)
If none of the angles of a triangle is a right angle, the triangle is called _____.(a) oblique(b) obtuse(c) acute(d) scalene
Find the area of the right triangle whose legs are of length 3 and 4.
In Problems 49–74, establish each identity. sin (a + 3) + sin(a - 3) = 2 sin a cos 3
True or False The area of a triangle equals one-half the product of the lengths of two of its sides times the sine of their included angle.
In Problems 51–72, establish each identity. cos (20) 1 + sin(20) cot 0 - 1 cot + 1.
True or False The Law of Cosines states that the square of one side of a triangle equals the sum of the squares of the other two sides, minus twice their product.
If two angles of a triangle measure 48° and 93°, what is the measure of the third angle?(a) 132°(b) 77°(c) 42°(d) 39°
In Problems 59–63, use the information given about the angles α and β to find the exact value of: (a) sin(a + B) (e) sin(2a) (b) cos(a + 3) (f) cos(2/3) (c) sin(a - 3) В (g) sin sin 2 (d) tan(a + B) (h) cos 2
In Problems 51–72, establish each identity. sin ²0 cos²0 = 1 sin² (20)
True or False A special case of the Law of Cosines is the Pythagorean Theorem.
In Problems 51–72, establish each identity. 0 2 sec2. 2 1 + cos 0
In Problems 49–74, establish each identity. cos(a + 3) + cos(a 3) = 2 cosacos
True or False When two sides and an angle are given at least one triangle can be formed.
Heron’s Formula is used to find the area of triangles.(a) ASA(b) SAS(c) SSS(d) AAS
In Problems 59–63, use the information given about the angles α and β to find the exact value of: 3 tana = 3π ₁ T < a < ³; tan 3 = ¹,0
In Problems 51–72, establish each identity. csc 2 2 2 1 - cose
In Problems 49–74, establish each identity. sin (a + 3) sin a cos 3 = 1+ cotatan 3
In Problems 49–74, establish each identity. sin (a + 3) = cos a cos tana + tanß
Problems 61–70. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Solve: 27x−1 = 9x+5
In Problems 59–63, use the information given about the angles α and β to find the exact value of: seca = 2, EIN π 2 < a < 0; secß: 3, 37 < B < 2TT
In Problems 51–72, establish each identity. cot ²2 = 2 || secv + 1 secv - 1
In Problems 49–74, establish each identity. cos(a - B) sin a cos cota + tan 3
In Problems 49–74, establish each identity. cos(a + 3) cos a cos = 1- tan atan 3
Problems 61–70. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Find the exact value of 7 cos (csc-¹3).
Problems 61–70. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Find the inverse function f−1 ofFind the range of f and the domain and range of f−1. 1-1/2 ≤ x
In Problems 59–63, use the information given about the angles α and β to find the exact value of: sina = 2 -3, n
In Problems 51–72, establish each identity. tan; = csc v 2 cot v
In Problems 64–69, find the exact value of each expression. cos(sin-13- cos-11)
In Problems 51–72, establish each identity. cose = 1 tan 1+ tan² 2 2
Problems 61–70. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.For y = 5cos (4x − π), find the amplitude, the period, and the phase shift.
In Problems 49–74, establish each identity. sin (a + 3) sin (3) tana + tanß tan a - tan 3
In Problems 49–74, establish each identity. cos(a + 3) cos (a - B) 1 tanatan 3 1 + tanatan 3
Problems 61–70. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.Find the exact value of tan π 6A
In Problems 64–69, find the exact value of each expression. sin cos-1 5 13 - cos-1 5,
In Problems 64–69, find the exact value of each expression. tan[sin-¹(-2) – tan-¹³]
In Problems 51–72, establish each identity. 1 - =sin(20) 2 = sin ³0+cos³0 sin 0 + cos 0
In Problems 51–72, establish each identity. cos + sin cos sin cos- sin cos + sin = 2 tan (20)
In Problems 51–72, establish each identity. sin (30) sine cos (30) cos
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