Stewart Precalc 6e Section 7.5: More Trigonometric Equations

1 Concept - Pythagorean Identity · Level 1

We can use identities to help us solve trigonometric equations. Using a Pythagorean identity we see that the equation \(\sin x + \sin^2 x + \cos^2 x = 1\) is equivalent to the basic equation ______ whose solutions are \(x = \) ______.

2 Concept - Double-Angle Formula · Level 1

Using a Double-Angle Formula we see that the equation \(\sin x + \sin 2 x = 0\) is equivalent to the equation ______. Factoring we see that solving this equation is equivalent to solving the two basic equations ______ and ______.

3 Solve - Pythagorean Identity · Level 3

Solve the given equation. \(2 \cos^2 \theta + \sin \theta = 1\)

4 Solve - Pythagorean Identity · Level 2

Solve the given equation. \(\sin^2 \theta = 4 - 2 \cos^2 \theta\)

5 Solve - Pythagorean Identity · Level 3

Solve the given equation. \(\tan^2 \theta - 2 \sec \theta = 2\)

6 Solve - Pythagorean Identity · Level 3

Solve the given equation. \(\csc^2 \theta = \cot \theta + 3\)

7 Solve - Double-Angle Formula · Level 3

Solve the given equation. \(2 \sin 2 \theta - 3 \sin \theta = 0\)

8 Solve - Double-Angle Formula · Level 3

Solve the given equation. \(3 \sin 2 \theta - 2 \sin \theta = 0\)

9 Solve - Double-Angle Formula · Level 3

Solve the given equation. \(\cos 2 \theta = 3 \sin \theta - 1\)

10 Solve - Double-Angle Formula · Level 3

Solve the given equation. \(\cos 2 \theta = \cos^2 \theta - \dfrac{1}{2}\)

11 Solve - Pythagorean Identity · Level 3

Solve the given equation. \(2 \sin^2 \theta - \cos \theta = 1\)

12 Solve - Trig Identities · Level 2

Solve the given equation. \(\tan \theta - 3 \cot \theta = 0\)

13 Solve - Squaring (check extraneous) · Level 4

Solve the given equation. \(\sin \theta - 1 = \cos \theta\)

14 Solve - Squaring (check extraneous) · Level 4

Solve the given equation. \(\cos \theta - \sin \theta = 1\)

15 Solve - Squaring (check extraneous) · Level 4

Solve the given equation. \(\tan \theta + 1 = \sec \theta\)

16 Solve - Pythagorean Identity · Level 3

Solve the given equation. \(2 \tan \theta + \sec^2 \theta = 4\)

17 Multiple Angle - Cosine · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(2 \cos 3 \theta = 1\)

18 Cosecant Equation · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(3 \csc^2 \theta = 4\)

19 Multiple Angle - Cosine · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(2 \cos 2 \theta + 1 = 0\)

20 Multiple Angle - Sine · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(2 \sin 3 \theta + 1 = 0\)

21 Multiple Angle - Tangent · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(\sqrt{3} \tan 3 \theta + 1 = 0\)

22 Multiple Angle - Secant · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(\sec 4 \theta - 2 = 0\)

23 Half Angle - Cosine · Level 2

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(\cos\left(\dfrac{\theta}{2}\right) - 1 = 0\)

24 Quarter Angle - Tangent · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(\tan\left(\dfrac{\theta}{4}\right) + \sqrt{3} = 0\)

25 Half Angle - Sine · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(2 \sin\left(\dfrac{\theta}{2}\right) + \sqrt{3} = 0\)

26 Half Angle - Reciprocal Identity · Level 3

An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\). \(\sec\left(\dfrac{\theta}{2}\right) = \cos\left(\dfrac{\theta}{2}\right)\)

27 Solving Trigonometric Equations · Level 3

Find all solutions in \([0, 2 \pi)\): \(\csc 3 \theta = 5 \sin 3 \theta\).

28 Solving Trigonometric Equations · Level 3

Find all solutions in \([0, 2 \pi)\): \(\sec \theta - \tan \theta = \cos \theta\).

29 Solving Trigonometric Equations · Level 3

Find all solutions in \([0, 2 \pi)\): \(\tan 3 \theta + 1 = \sec 3 \theta\).

30 Solving Trigonometric Equations · Level 3

Find all solutions in \([0, 2 \pi)\): \(3 \tan^3 \theta - 3 \tan^2 \theta - \tan \theta + 1 = 0\).

31 Solving Trigonometric Equations · Level 3

Find all solutions in \([0, 2 \pi)\): \(4 \sin \theta \cos \theta + 2 \sin \theta - 2 \cos \theta - 1 = 0\).

32 Solving Trigonometric Equations · Level 3

Find all solutions in \([0, 2 \pi)\): \(2 \sin \theta \tan \theta - \tan \theta = 1 - 2 \sin \theta\).

33 Solving Trigonometric Equations · Level 3

Find all solutions in \([0, 2 \pi)\): \(\sec \theta \tan \theta - \cos \theta \cot \theta = \sin \theta\).

34 Graphical and Algebraic Intersection · Level 2

(a) Graph \(f(x) = 3 \cos x + 1\) and \(g(x) = \cos x - 1\) in the viewing rectangle \([-2 \pi, 2 \pi]\) by \([-2.5, 4.5]\) and find the intersection points graphically (rounded to two decimal places). (b) Find the intersection points algebraically; give exact values.

35 Graphical and Algebraic Intersection · Level 2

(a) Graph \(f(x) = \sin 2 x + 1\) and \(g(x) = 2 \sin 2 x + 1\) in the viewing rectangle \([-2 \pi, 2 \pi]\) by \([-1.5, 3.5]\) and find intersection points graphically. (b) Find them algebraically.

36 Graphical and Algebraic Intersection · Level 2

(a) Graph \(f(x) = \tan x\) and \(g(x) = \sqrt{3}\) in the viewing rectangle \([-\dfrac{\pi}{2}, \dfrac{\pi}{2}]\) by \([-10, 10]\) and find intersection points graphically. (b) Find them algebraically.

37 Graphical and Algebraic Intersection · Level 3

(a) Graph \(f(x) = \sin x - 1\) and \(g(x) = \cos x\) in the viewing rectangle \([-2 \pi, 2 \pi]\) by \([-2.5, 1.5]\) and find intersection points graphically. (b) Find them algebraically.

38 Addition and Subtraction Formulas · Level 2

Use an Addition or Subtraction Formula to simplify, then find all solutions in \([0, 2 \pi)\): \(\cos \theta \cos 3 \theta - \sin \theta \sin 3 \theta = 0\).

39 Addition and Subtraction Formulas · Level 2

Use an Addition or Subtraction Formula to simplify, then find all solutions in \([0, 2 \pi)\): \(\cos \theta \cos 2 \theta + \sin \theta \sin 2 \theta = \dfrac{1}{2}\).

40 Addition and Subtraction Formulas · Level 2

Use an Addition or Subtraction Formula to simplify, then find all solutions in \([0, 2 \pi)\): \(\sin 2 \theta \cos \theta - \cos 2 \theta \sin \theta = \dfrac{\sqrt{3}}{2}\).

41 Addition and Subtraction Formulas · Level 2

Use an Addition or Subtraction Formula to simplify, then find all solutions in \([0, 2 \pi)\): \(\sin 3 \theta \cos \theta - \cos 3 \theta \sin \theta = 0\).

42 Double- or Half-Angle Formulas · Level 2

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\sin 2 \theta + \cos \theta = 0\).

43 Double- or Half-Angle Formulas · Level 3

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\tan\left(\dfrac{\theta}{2}\right) - \sin \theta = 0\).

44 Double- or Half-Angle Formulas · Level 2

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\cos 2 \theta + \cos \theta = 2\).

45 Double- or Half-Angle Formulas · Level 3

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\tan \theta + \cot \theta = 4 \sin 2 \theta\).

46 Double- or Half-Angle Formulas · Level 2

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\cos 2 \theta - \cos^2 \theta = 0\).

47 Double- or Half-Angle Formulas · Level 2

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(2 \sin^2 \theta = 2 + \cos 2 \theta\).

48 Double- or Half-Angle Formulas · Level 3

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\cos 2 \theta - \cos 4 \theta = 0\).

49 Double- or Half-Angle Formulas · Level 3

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\sin 3 \theta - \sin 6 \theta = 0\).

50 Double- or Half-Angle Formulas · Level 4

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\cos \theta - \sin \theta = \sqrt{2} \sin\left(\dfrac{\theta}{2}\right)\).

51 Double- or Half-Angle Formulas · Level 3

Use a Double- or Half-Angle Formula to solve in \([0, 2 \pi)\): \(\sin \theta - \cos \theta = \dfrac{1}{2}\).

52 Sum-to-Product · Level 2

Solve by first using a Sum-to-Product Formula in \([0, 2 \pi)\): \(\sin \theta + \sin 3 \theta = 0\).

53 Sum-to-Product · Level 2

Solve by first using a Sum-to-Product Formula in \([0, 2 \pi)\): \(\cos 5 \theta - \cos 7 \theta = 0\).

54 Sum-to-Product · Level 3

Solve by first using a Sum-to-Product Formula in \([0, 2 \pi)\): \(\cos 4 \theta + \cos 2 \theta = \cos \theta\).

55 Sum-to-Product · Level 3

Solve by first using a Sum-to-Product Formula in \([0, 2 \pi)\): \(\sin 5 \theta - \sin 3 \theta = \cos 4 \theta\).

56 Graphing Device · Level 2

Use a graphing device to find solutions correct to two decimal places: \(\sin 2 x = x\).

57 Graphing Device · Level 2

Use a graphing device to find solutions correct to two decimal places: \(\cos x = \dfrac{x}{3}\).

58 Graphing Device · Level 2

Use a graphing device to find solutions correct to two decimal places: \(2^{\sin x} = x\).

59 Graphing Device · Level 2

Use a graphing device to find solutions correct to two decimal places: \(\sin x = x^3\).

60 Graphing Device · Level 2

Use a graphing device to find solutions correct to two decimal places: \(\cos x/(1 + x^2) = x^2\).

61 Graphing Device · Level 2

Use a graphing device to find solutions correct to two decimal places: \(\cos x = \left(\dfrac{1}{2}\right)(e^x + e^{-x})\).

62 Applications · Level 3

Range of a Projectile. If a projectile is fired with velocity \(v_0\) at angle \(\theta\), its range (in feet) is \(R(\theta) = (v_0^2 \sin 2 \theta)/32\). If \(v_0 = 2200\) ft/s, what angle (in degrees) should be chosen for the projectile to hit a target 5000 ft away?

63 Applications · Level 2

Damped Vibrations. The displacement of a spring vibrating in damped harmonic motion is \(y = 4 e^{-3 t} \sin 2 \pi t\). Find the times when the spring is at its equilibrium position \((y = 0)\).

64 Applications · Level 3

Hours of Daylight. In Philadelphia, the number of hours of daylight on day \(t\) (where \(t\) is days after January
1) is \(L(t) = 12 + 2.83 \sin((2 \pi)/365 (t - 80))\). (a) Which days have about 10 hours of daylight? (b) How many days of the year have more than 10 hours of daylight?

65 Applications · Level 4

Belts and Pulleys. A thin belt of length \(L\) surrounds two pulleys of radii \(R\) and \(r\), as shown in the figure. (a) Show that the angle \(\theta\) (in radians) where the belt crosses itself satisfies \(\theta + 2 \cot\left(\dfrac{\theta}{2}\right) = L/(R + r) - \pi\). (b) Suppose \(R = 2.42\) ft, \(r = 1.21\) ft, \(L = 27.78\) ft. Find \(\theta\) by solving graphically; express in radians and degrees.

66 Discovery and Discussion · Level 3

A Special Trigonometric Equation. What makes the equation \(\sin(\cos x) = 0\) different from the other equations in this section? Find all solutions of this equation.

67 Example - Using a Trigonometric Identity · Level 3

Solve the equation \(1 + \sin \theta = 2 \cos^2 \theta\).

68 Example - Using a Trigonometric Identity · Level 3

Solve the equation \(\sin 2 \theta - \cos \theta = 0\).

69 Example - Squaring and Using an Identity · Level 3

Solve the equation \(\cos \theta + 1 = \sin \theta\) in the interval \([0, 2 \pi)\).

70 Example - Finding Intersection Points · Level 3

Find the values of \(x\) for which the graphs of \(f(x) = \sin x\) and \(g(x) = \cos x\) intersect.

71 Example - Trigonometric Equation Involving a Multiple of an Angle · Level 3

Consider the equation \(2 \sin 3 \theta - 1 = 0\). (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 2 \pi)\).

72 Example - Trigonometric Equation Involving a Half Angle · Level 3

Consider the equation \(\sqrt{3} \tan\left(\dfrac{\theta}{2}\right) - 1 = 0\). (a) Find all solutions of the equation. (b) Find the solutions in the interval \([0, 4 \pi)\).

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