Using the inverse trigonometric functions, we can write down general solutions to trigonometric equations of the type sinx = a and sin x = sin α.
Solving Trigonometric Equations Without Restrictions: Because each of the trigono-metric functions is periodic, any unrestricted trigonotrigono-metric equation that has one solution must have infinitely many. This section will later develop formulae for those general solutions, but they can always be found using the methods al-ready established, as is demonstrated in the following worked exercise. The key to general solutions is provided by the periods of the trigonometric functions:
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PERIODS OF THE TRIGONMETRIC FUNCTIONS:
sinx and cos x have period 2π, tanx has period π.
WORKEDEXERCISE: Find the general solution, in radians, of:
(a) cosx = 12 (b) tanx = 1 (c) sinx = 12√ 3 SOLUTION:
(a) Since cosx is positive, x must be in the 1st or 4th quadrants.
Also, the related acute angle is π3.
Hence x = π3 and x = −π3 are the solutions within a revolution.
x
x
π3 π3
Since cosx has period 2π, the general solution is x = π3 + 2nπ or −π3 + 2nπ, where n is an integer.
(b) Since tanx is positive, x must be in the 1st or 3rd quadrants.
Also, the related angle is π4.
Hence x = π4 and x = 5π4 are the solutions within a revolution.
Since tanx has period π, the general solution is
π4 π4
x
x = π4 +nπ, where n is an integer. x
(Notice that this includes the other solution x = 5π4 , which is obtained by putting n = 1.)
(c) Since sinx is positive, x must be in the 1st or 2nd quadrants.
Also, the related acute angle is π3.
Hence x = π3 and x = π − π3 = 23π are both solutions.
Since sinx has period 2π, the general solution is
π3 π3
x x
x = π3 + 2nπ or 23π + 2nπ, where n is an integer.
The Equation cosx = a: More generally, suppose that cosx = a, where −1 ≤ a ≤ 1.
First,x = cos−1a is a solution.
Secondly,x = − cos−1a is a solution, because cos x is an even function.
This gives two solutions within a revolution, so the general solution is x = cos−1a + 2nπ or x = − cos−1a + 2nπ, where n is an integer.
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THE GENERAL SOLUTION OFcosx = a: The general solution of cos x = a is x = cos−1a + 2nπ or x = − cos−1a + 2nπ, where n is an integer.
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CHAPTER1: The Inverse Trigonometric Functions 1F General Solutions of Trigonometric Equations 33r
The Equation tanx = a: Suppose that tanx = a, where a is a constant.
One solution is x = tan−1a.
But tanx has period π, and only one solution within each period, so the general solution isx = tan−1a + nπ, where n is an integer.
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THE GENERAL SOLUTION OFtanx = a: The general solution of tan x = a is x = tan−1a + nπ, where n is an integer.
The Equation sinx = a: Suppose that sinx = a, where −1 ≤ a ≤ 1.
First,x = sin−1a is a solution.
Also, if sinθ = a, then sin(π − θ) = a, so x = π − sin−1a is a solution.
This gives two solutions within each revolution, so the general solution is x = sin−1a + 2nπ or x = (π − sin−1a) + 2nπ, where n is an integer.
[Alternatively, we can write x = 2nπ + sin−1a or x = (2n + 1)π − sin−1a.
The first can be written asx = mπ + sin−1a, where m is even, and the second can be written asx = mπ − sin−1a, where m is odd.
Using the switch (−1)m, which changes sign according as m is even or odd,
we can write both families together as x = (−1)m sin−1a + mπ, where m is an integer.]
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THE GENERAL SOLUTION OFsinx = a: The general solution of sin x = a is
x = sin−1a + 2nπ or x = (π − sin−1a) + 2nπ, where n is an integer.
[Alternatively, we can write these two families together using the switch (−1)m: x = (−1)msin−1a + mπ, where m is an integer.]
Note: The alternative notation for solving sin−1x = a is very elegant, and is very quick if properly applied, but it is not at all easy to use or to remember. In this text, we will enclose it in square brackets when it is used.
WORKEDEXERCISE: Use these formulae to find the general solution of:
(a) cosx = −12 (b) sinx = 12√
2 (c) tanx = −2
SOLUTION:
(a) x = cos−1(−12) + 2nπ or − cos−1(−12) + 2nπ, where n is an integer,
= 23π + 2nπ or −23π + 2nπ.
(b) x = sin−1 12√
2 + 2nπ or (π − sin−1 12√
2 ) + 2nπ, where n is an integer,
= π4 + 2nπ or 34π + 2nπ.
[Alternatively, x = (−1)m π4 +mπ, where m is an integer.]
(c) x = tan−1(−2) + nπ, where n is an integer,
=− tan−12 +nπ, which can be approximated if required.
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34 CHAPTER1: The Inverse Trigonometric Functions CAMBRIDGEMATHEMATICS3 UNITYEAR12r
The Equations sinx = sin α, cos x = cos α and tan x = tan α: Using similar meth-ods, the general solutions of these three equations can be written down.
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GENERAL SOLUTIONS OFsinx = sin α,cosx = cos αandtanx = tan α:
The general solution of cosx = cos α is
x = α + 2nπ or x = −α + 2nπ, where n is an integer.
The general solution of tanx = tan α is x = α + nπ, where n is an integer.
The general solution of sinx = sin α is
x = α + 2nπ or x = (π − α) + 2nπ, where n is an integer.
[Alternatively, x = mπ + (−1)mα, where m is an integer.]
Proof:
A. One solution of cosx = cos α is x = α.
Also, cosα = cos(−α), since cosine is even, so x = −α is also a solution.
This gives the required two solutions within a single period of 2π,
so the general solution is x = α + 2nπ or x = −α + 2nπ, where n is an integer.
B. One solution of tanx = tan α is x = α.
This gives the required one solution within a single period of π, so the general solution is x = α + nπ, where n is an integer.
C. One solution of sinx = sin α is x = α.
Also, sinα = sin(π − α), so x = π − α is also a solution.
This gives the required two solutions within a single period of 2π,
so the general solution is x = α + 2nπ or x = (π − α) + 2nπ, where n is an integer.
WORKEDEXERCISE: Use these formulae to find the general solution of sinx = sinπ5. SOLUTION: x = π5 + 2nπ or x = (π −π5) + 2nπ, where n is an integer,
x = π5 + 2nπ or x = 45π + 2nπ.
[Alternatively, x = (−1)m π5 +mπ, where n is an integer.]
WORKEDEXERCISE: Solve: (a) cos 4x = cos x (b) sin 4x = cos x SOLUTION:
(a) Using the general solution of cosx = cos α from Box 21,
4x = x + 2nπ or 4x = −x + 2nπ, where n ∈ Z, 3x = 2nπ or 5x = 2nπ, where n ∈ Z,
x = 23nπ or x = 25nπ, where n ∈ Z.
(b) First, sin 4x = sin(π2 − x), using the identity cos x = sin(π2 − x).
Hence, using the general solution of sinx = sin α from Box 21,
4x = π2 − x + 2nπ or 4x = π − (π2 − x) + 2nπ, where n ∈ Z, 5x = (2n +12)π or 4x = x +π2 + 2nπ, where n ∈ Z,
x = (4n + 1)10π or 3x = (2n +12)π, where n ∈ Z, x = (4n + 1)π6, where n ∈ Z.
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CHAPTER1: The Inverse Trigonometric Functions 1F General Solutions of Trigonometric Equations 35r
Exercise 1F
1. Consider the equation tanx = 1.
(a) Draw a diagram showingx in its two possible quadrants, and show the related angle.
(b) Write down the first six positive solutions.
(c) Write down the first six negative solutions.
(d) Carefully observe that each of these twelve solutions can be written as an integer multiple ofπ plus π4, and hence write down a general solution of tanx = 1.
(e) Sketch the graphs of y = tan x (for −2π ≤ x ≤ 2π) and y = 1 on the same diagram and show as many of the above solutions as possible.
2. Consider the equation cosx = 12.
(a) Draw a diagram showingx in its two possible quadrants, and show the related angle.
(b) Write down the first six positive solutions.
(c) Write down the first six negative solutions.
(d) Carefully observe that each of these twelve solutions can be written either as an integer multiple of 2π plus π3 or as an integer multiple of 2π minus π3, and hence write down a general solution of cosx = 12.
(e) Sketch the graphs of y = cos x (for −2π ≤ x ≤ 2π) and y = 12 on the same diagram and show as many of the above solutions as possible.
3. Consider the equation sinx = 12.
(a) Draw a diagram showingx in its two possible quadrants, and show the related angle.
(b) Write down the first six positive solutions.
(c) Write down the first six negative solutions.
(d) Carefully observe that each of these twelve solutions can be written either as a multiple of 2π plus π6 or as a multiple of 2π plus 5π6 , and hence write down a general solution of sinx = 12.
(e) Sketch the graphs of y = sin x (for −2π ≤ x ≤ 2π) and y = 12 on the same diagram and show as many of the above solutions as possible.
4. Write down a general solution of:
(a) tanx =√ 5. Write down a general solution of:
(a) cosθ = cosπ6
6. Write down a general solution for each of the following by referring to the graphs of y = sin x, y = cos x and y = tan x.
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36 CHAPTER1: The Inverse Trigonometric Functions CAMBRIDGEMATHEMATICS3 UNITYEAR12r
8. In each case: (i) find a general solution, (ii) write down all solutions in−π ≤ θ ≤ π.
(a) sin2θ + sin θ = 0 (b) sin 2θ = cos θ
(c) cot2(θ − π6) = 3 (d) 2 sin2θ = 3 + 3 cos θ
(e) sin 2θ +√
3 cos 2θ = 0 (f) sec22θ = 1 + tan 2θ 9. Consider the equation tan 4x = tan x.
(a) Show that 4x = nπ + x. (b) Hence show that x = nπ3 , wheren ∈ Z.
(c) Hence write down all solutions in the domain 0≤ x ≤ 2π.
10. Consider the equation sin 3x = sin x.
(a) Show that 3x = x + 2nπ or 3x = (π − x) + 2nπ. Hence show that x = nπ or x = (2n + 1)π4. [Alternatively, show that 3x = nπ + (−1)nx, and hence show that if n is even, x = nπ2 , and if n is odd, x = nπ4 .]
(b) Hence write down all solutions in the domain 0≤ x ≤ 2π.
11. Consider the equation cos 3x = sin x.
(a) Show that 3x = 2nπ + (π2 − x) or 2nπ − (π2 − x).
(b) Hence show that x = nπ − π4 or nπ2 +π8, wheren ∈ Z.
(c) Hence write down all solutions in the domain 0≤ x ≤ 2π.
12. Using methods similar to those in the previous two questions, solve for 0≤ x ≤ π:
(a) sin 5x = sin x (b) cos 5x = cos x
(c) sin 5x = cos x (d) cos 5x = sin x
E X T E N S I O N
13. Sketch the graphs of the following relations:
(a) cosy = cos x (b) siny = sin x
(c) cosy = sin x (d) tany = tan x
(e) coty = tan x (f) secy = sec x Which graphs are symmetric in thex-axis, which in the y-axis, and which in y = x?