Stewart Precalc 6e Section 7.4: Basic Trigonometric Equations

1 Concept - Periodic Solutions · Level 1

Because the trigonometric functions are periodic, if a basic trigonometric equation has one solution, it has _____ (several/infinitely many) solutions.

2 Concept - Existence of Solutions · Level 1

The basic equation \(\sin x = 2\) has _____ (no/one/infinitely many) solutions, whereas the basic equation \(\sin x = 0.3\) has _____ (no/one/infinitely many) solutions.

3 Concept - Graphical Solution · Level 1

We can find some of the solutions of \(\sin x = 0.3\) graphically by graphing \(y = \sin x\) and \(y =\) _____. Use the graph below to estimate some of the solutions.

4 Concept - Algebraic Solution · Level 1

We can find the solutions of \(\sin x = 0.3\) algebraically.

(a) First we find the solutions in the interval \([-\pi, \pi]\). We get one such solution by taking \(\sin^{-1}\) to get \(x =\) _____. The other solution in this interval is \(x =\) _____.

Answer for 4(a)

(b) We find all solutions by adding multiples of _____ to the solutions in \([-\pi, \pi]\). The solutions are \(x =\) _____ and \(x =\) _____.

Answer for 4(b)

Enter your answer directly below each part above.

5 Skills - Basic Sine Equation · Level 2

Solve the equation \(\sin \theta = \dfrac{\sqrt{3}}{2}\).

6 Skills - Basic Sine Equation · Level 2

Solve the equation \(\sin \theta = -\dfrac{\sqrt{2}}{2}\).

7 Skills - Basic Cosine Equation · Level 2

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

8 Skills - Basic Cosine Equation · Level 2

Solve the equation \(\cos \theta = \dfrac{\sqrt{3}}{2}\).

9 Skills - Basic Cosine Equation · Level 2

Solve the equation \(\cos \theta = \dfrac{1}{4}\).

10 Skills - Basic Sine Equation · Level 2

Solve the equation \(\sin \theta = -0.3\).

11 Skills - Basic Sine Equation · Level 2

Solve the equation \(\sin \theta = -0.45\).

12 Skills - Basic Cosine Equation · Level 2

Solve the equation \(\cos \theta = 0.32\).

13 Skills - Basic Tangent Equation · Level 2

Solve the equation \(\tan \theta = -\sqrt{3}\).

14 Skills - Basic Tangent Equation · Level 2

Solve the equation \(\tan \theta = 1\).

15 Skills - Basic Tangent Equation · Level 2

Solve the equation \(\tan \theta = 5\).

16 Skills - Basic Tangent Equation · Level 2

Solve the equation \(\tan \theta = -\dfrac{1}{3}\).

17 Skills - List Specific Solutions · Level 2

Solve the equation \(\cos \theta = -\dfrac{\sqrt{3}}{2}\), and list six specific solutions.

18 Skills - List Specific Solutions · Level 2

Solve the equation \(\cos \theta = \dfrac{1}{2}\), and list six specific solutions.

19 Skills - List Specific Solutions · Level 2

Solve the equation \(\sin \theta = \dfrac{\sqrt{2}}{2}\), and list six specific solutions.

20 Skills - List Specific Solutions · Level 2

Solve the equation \(\sin \theta = -\dfrac{\sqrt{3}}{2}\), and list six specific solutions.

21 Skills - List Specific Solutions · Level 2

Solve the equation \(\cos \theta = 0.28\), and list six specific solutions.

22 Skills - List Specific Solutions · Level 2

Solve the equation \(\tan \theta = 2.5\), and list six specific solutions.

23 Skills - List Specific Solutions · Level 2

Solve the equation \(\tan \theta = -10\), and list six specific solutions.

24 Skills - List Specific Solutions · Level 2

Solve the equation \(\sin \theta = -0.9\), and list six specific solutions.

25 Skills - Find All Solutions · Level 2

Find all solutions of the equation \(\cos \theta + 1 = 0\).

26 Skills - Find All Solutions · Level 2

Find all solutions of the equation \(\sin \theta + 1 = 0\).

27 Skills - Find All Solutions · Level 2

Find all solutions of the equation \(\sqrt{2} \sin \theta + 1 = 0\).

28 Skills - Find All Solutions · Level 2

Find all solutions of the equation \(\sqrt{2} \cos \theta - 1 = 0\).

29 Skills - Find All Solutions · Level 2

Find all solutions of the equation \(5 \sin \theta - 1 = 0\).

30 Skills - Find All Solutions · Level 2

Find all solutions of the equation \(4 \cos \theta + 1 = 0\).

31 Skills - Find All Solutions · Level 3

Find all solutions of the equation \(3 \tan^2 \theta - 1 = 0\).

32 Skills - Find All Solutions · Level 3

Find all solutions of the equation \(\cot \theta + 1 = 0\).

33 Skills - Find All Solutions · Level 3

Find all solutions of the equation \(2 \cos^2 \theta - 1 = 0\).

34 Skills - Find All Solutions · Level 3

Find all solutions of the equation \(4 \sin^2 \theta - 3 = 0\).

35 Skills - Find All Solutions · Level 3

Find all solutions of the equation \(\tan^2 \theta - 4 = 0\).

36 Skills - Find All Solutions · Level 3

Find all solutions of the equation \(9 \sin^2 \theta - 1 = 0\).

37 Skills - Find All Solutions · Level 3

Find all solutions of the equation \(\sec^2 \theta - 2 = 0\).

38 Skills - Find All Solutions · Level 3

Find all solutions of the equation \(\csc^2 \theta - 4 = 0\).

39 Skills - Factoring · Level 3

Solve the equation \((\tan^2 \theta - 4)(2 \cos \theta + 1) = 0\).

40 Skills - Factoring · Level 3

Solve the equation \((\tan \theta - 2)(16 \sin^2 \theta - 1) = 0\).

41 Skills - Factoring (Perfect Square) · Level 3

Solve the equation \(4 \cos^2 \theta - 4 \cos \theta + 1 = 0\).

42 Skills - Factoring · Level 3

Solve the equation \(2 \sin^2 \theta - \sin \theta - 1 = 0\).

43 Skills - Factoring · Level 3

Solve the equation \(3 \sin^2 \theta - 7 \sin \theta + 2 = 0\).

44 Skills - Factoring · Level 3

Solve the equation \(\tan^4 \theta - 13 \tan^2 \theta + 36 = 0\).

45 Skills - Factoring · Level 3

Solve the equation \(2 \cos^2 \theta - 7 \cos \theta + 3 = 0\).

46 Skills - Factoring · Level 3

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

47 Skills - Factoring · Level 3

Solve the equation \(\cos^2 \theta - \cos \theta - 6 = 0\).

48 Skills - Factoring · Level 3

Solve the equation \(2 \sin^2 \theta + 5 \sin \theta - 12 = 0\).

49 Skills - Factoring · Level 3

Solve the equation \(\sin^2 \theta = 2 \sin \theta + 3\).

50 Skills - Factoring · Level 3

Solve the equation \(3 \tan^3 \theta = \tan \theta\).

51 Skills - Factoring · Level 3

Solve the equation \(\cos \theta (2 \sin \theta + 1) = 0\).

52 Skills - Factoring · Level 3

Solve the equation \(\sec \theta (2 \cos \theta - \sqrt{2}) = 0\).

53 Skills - Factoring · Level 3

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

54 Skills - Factoring · Level 3

Solve the equation \(\tan \theta \sin \theta + \sin \theta = 0\).

55 Skills - Factoring · Level 3

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

56 Skills - Factoring · Level 3

Solve the equation \(4 \cos \theta \sin \theta + 3 \cos \theta = 0\).

57 Application - Refraction of Light · Level 3

Refraction of Light. It has been observed since ancient times that light refracts or bends as it travels from one medium to another (from air to water, for example). If \(v_1\) is the speed of light in one medium and \(v_2\) its speed in another medium, then according to Snell's Law, \(\dfrac{\sin \theta_1}{\sin \theta_2} = \dfrac{v_1}{v_2}\) where \(\theta_1\) is the angle of incidence and \(\theta_2\) is the angle of refraction (see the figure). The number \(\dfrac{v_1}{v_2}\) is called the index of refraction. The index of refraction for several substances is given in the table (Water: 1.33, Alcohol: 1.36, Glass: 1.52, Diamond: 2.41). If a ray of light passes through the surface of a lake at an angle of incidence of \(70^{\circ}\), what is the angle of refraction?

58 Application - Total Internal Reflection · Level 3

Total Internal Reflection. When light passes from a more-dense to a less-dense medium — from glass to air, for example — the angle of refraction predicted by Snell's Law (see Exercise 57) can be \(90^{\circ}\) or larger. In this case the light beam is actually reflected back into the denser medium. This phenomenon, called total internal reflection, is the principle behind fiber optics. Set \(\theta_2 = 90^{\circ}\) in Snell's Law, and solve for \(\theta_1\) to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. (Note that the index of refraction from glass to air is the reciprocal of the index from air to glass.)

59 Application - Phases of the Moon · Level 3

Phases of the Moon. As the moon revolves around the earth, the side that faces the earth is usually just partially illuminated by the sun. The phases of the moon describe how much of the surface appears to be in sunlight. An astronomical measure of phase is given by the fraction \(F\) of the lunar disc that is lit. When the angle between the sun, earth, and moon is \(\theta\) \((0 \leq \theta \leq 360^{\circ})\), then \(F = \dfrac{1}{2}(1 - \cos \theta)\) Determine the angles \(\theta\) that correspond to the following phases:

(a) \(F = 0\) (new moon)

Answer for 59(a)

(b) \(F = 0.25\) (a crescent moon)

Answer for 59(b)

(c) \(F = 0.5\) (first or last quarter)

Answer for 59(c)

(d) \(F = 1\) (full moon)

Answer for 59(d)

Enter your answer directly below each part above.

60 Discussion - Equations and Identities · Level 2

Equations and Identities. Which of the following statements is true? A. Every identity is an equation. B. Every equation is an identity. Give examples to illustrate your answer. Write a short paragraph to explain the difference between an equation and an identity.