Solving simple trigonometric equations.
Solving trigonometric equations of any level of complexity ultimately comes down to solving the simplest trigonometric equations. And in this best helper again it turns out to be a trigonometric circle.
Let's recall the definitions of cosine and sine.
The cosine of an angle is the abscissa (that is, the coordinate along the axis) of a point on unit circle, corresponding to rotation through a given angle.
The sine of an angle is the ordinate (that is, the coordinate along the axis) of a point on the unit circle corresponding to a rotation through a given angle.
Positive direction of movement along trigonometric circle Counterclockwise movement is considered. A rotation of 0 degrees (or 0 radians) corresponds to a point with coordinates (1;0)
We use these definitions to solve simple trigonometric equations.
1. Solve the equation
This equation is satisfied by all values of the rotation angle that correspond to points on the circle whose ordinate is equal to .
Let's mark a point with ordinate on the ordinate axis:
Draw a horizontal line parallel to the x-axis until it intersects with the circle. We get two points lying on the circle and having an ordinate. These points correspond to rotation angles in and radians:
If we, leaving the point corresponding to the angle of rotation per radian, go around a full circle, then we will arrive at a point corresponding to the angle of rotation per radian and having the same ordinate. That is, this rotation angle also satisfies our equation. We can make as many “idle” revolutions as we like, returning to the same point, and all these angle values will satisfy our equation. The number of “idle” revolutions will be denoted by the letter (or). Since we can make these revolutions in both positive and negative directions, (or) can take on any integer values.
That is, the first series of solutions to the original equation has the form:
, , - set of integers (1)
Similarly, the second series of solutions has the form:
, Where , . (2)
As you might have guessed, this series of solutions is based on the point on the circle corresponding to the angle of rotation by .
These two series of solutions can be combined into one entry:
If we take (that is, even) in this entry, then we will get the first series of solutions.
If we take (that is, odd) in this entry, then we get the second series of solutions.
2. Now let's solve the equation
Since this is the abscissa of a point on the unit circle obtained by rotating through an angle, we mark the point with the abscissa on the axis:
Draw a vertical line parallel to the axis until it intersects with the circle. We will get two points lying on the circle and having an abscissa. These points correspond to rotation angles in and radians. Recall that when moving clockwise we get a negative rotation angle:
Let us write down two series of solutions:
,
,
(We get to the desired point by going from the main full circle, that is.
Let's combine these two series into one entry:
3. Solve the equation
The tangent line passes through the point with coordinates (1,0) of the unit circle parallel to the OY axis
Let's mark a point on it with an ordinate equal to 1 (we are looking for the tangent of which angles is equal to 1):
Let's connect this point to the origin of coordinates with a straight line and mark the points of intersection of the line with the unit circle. The intersection points of the straight line and the circle correspond to the angles of rotation on and:
Since the points corresponding to the rotation angles that satisfy our equation lie at a distance of radians from each other, we can write the solution this way:
4. Solve the equation
The line of cotangents passes through the point with the coordinates of the unit circle parallel to the axis.
Let's mark a point with abscissa -1 on the cotangent line:
Let's connect this point to the origin of the straight line and continue it until it intersects with the circle. This straight line will intersect the circle at points corresponding to the angles of rotation in and radians:
Since these points are separated from each other by a distance equal to , then general solution We can write this equation like this:
In the given examples illustrating the solution of the simplest trigonometric equations, tabular values of trigonometric functions were used.
However, if the right side of the equation contains a non-tabular value, then we substitute the value into the general solution of the equation:
SPECIAL SOLUTIONS:
Let us mark the points on the circle whose ordinate is 0:
Let us mark a single point on the circle whose ordinate is 1:
Let us mark a single point on the circle whose ordinate is equal to -1:
Since it is customary to indicate values closest to zero, we write the solution as follows:
Let us mark the points on the circle whose abscissa is equal to 0:
5.
Let us mark a single point on the circle whose abscissa is equal to 1:
Let us mark a single point on the circle whose abscissa is equal to -1:
And slightly more complex examples:
1.
The sine is equal to one if the argument is equal to
The argument of our sine is equal, so we get:
Let's divide both sides of the equality by 3:
Answer:
2.
Cosine is zero if the argument of cosine is
The argument of our cosine is equal to , so we get:
Let's express , to do this we first move to the right with the opposite sign:
Let's simplify the right side:
Divide both sides by -2:
Note that the sign in front of the term does not change, since k can take any integer value.
Answer:
And finally, watch the video tutorial “Selecting roots in a trigonometric equation using trigonometric circle"
This concludes our conversation about solving simple trigonometric equations. Next time we will talk about how to decide.
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The simplest trigonometric equations are solved, as a rule, using formulas. Let me remind you that the simplest trigonometric equations are:
sinx = a
cosx = a
tgx = a
ctgx = a
x is the angle to be found,
a is any number.
And here are the formulas with which you can immediately write down the solutions to these simplest equations.
For sine:
For cosine:
x = ± arccos a + 2π n, n ∈ Z
For tangent:
x = arctan a + π n, n ∈ Z
For cotangent:
x = arcctg a + π n, n ∈ Z
Actually, this is the theoretical part of solving the simplest trigonometric equations. Moreover, everything!) Nothing at all. However, the number of errors on this topic is simply off the charts. Especially if the example deviates slightly from the template. Why?
Yes, because a lot of people write down these letters, without understanding their meaning at all! He writes down with caution, lest something happen...) This needs to be sorted out. Trigonometry for people, or people for trigonometry, after all!?)
Let's figure it out?
One angle will be equal to arccos a, second: -arccos a.
And it will always work out this way. For any A.
If you don’t believe me, hover your mouse over the picture, or touch the picture on your tablet.) I changed the number A to something negative. Anyway, we got one corner arccos a, second: -arccos a.
Therefore, the answer can always be written as two series of roots:
x 1 = arccos a + 2π n, n ∈ Z
x 2 = - arccos a + 2π n, n ∈ Z
Let's combine these two series into one:
x= ± arccos a + 2π n, n ∈ Z
And that's all. We have obtained a general formula for solving the simplest trigonometric equation with cosine.
If you understand that this is not some kind of superscientific wisdom, but just a shortened version of two series of answers, You will also be able to handle tasks “C”. With inequalities, with selecting roots from a given interval... There the answer with a plus/minus does not work. But if you treat the answer in a businesslike manner and break it down into two separate answers, everything will be resolved.) Actually, that’s why we’re looking into it. What, how and where.
In the simplest trigonometric equation
sinx = a
we also get two series of roots. Always. And these two series can also be recorded in one line. Only this line will be trickier:
x = (-1) n arcsin a + π n, n ∈ Z
But the essence remains the same. Mathematicians simply designed a formula to make one instead of two entries for series of roots. That's all!
Let's check the mathematicians? And you never know...)
In the previous lesson, the solution (without any formulas) of a trigonometric equation with sine was discussed in detail:
The answer resulted in two series of roots:
x 1 = π /6 + 2π n, n ∈ Z
x 2 = 5π /6 + 2π n, n ∈ Z
If we solve the same equation using the formula, we get the answer:
x = (-1) n arcsin 0.5 + π n, n ∈ Z
Actually, this is an unfinished answer.) The student must know that arcsin 0.5 = π /6. The complete answer would be:
x = (-1)n π /6+ π n, n ∈ Z
Here it arises interesting question. Reply via x 1; x 2 (this is the correct answer!) and through lonely X (and this is the correct answer!) - are they the same thing or not? We'll find out now.)
We substitute in the answer with x 1 values n =0; 1; 2; etc., we count, we get a series of roots:
x 1 = π/6; 13π/6; 25π/6 and so on.
With the same substitution in response with x 2 , we get:
x 2 = 5π/6; 17π/6; 29π/6 and so on.
Now let's substitute the values n (0; 1; 2; 3; 4...) into the general formula for single X . That is, we raise minus one to the zero power, then to the first, second, etc. Well, of course, we substitute 0 into the second term; 1; 2 3; 4, etc. And we count. We get the series:
x = π/6; 5π/6; 13π/6; 17π/6; 25π/6 and so on.
That's all you can see.) The general formula gives us exactly the same results as are the two answers separately. Just everything at once, in order. The mathematicians were not fooled.)
Formulas for solving trigonometric equations with tangent and cotangent can also be checked. But we won’t.) They are already simple.
I wrote out all this substitution and checking specifically. It is important to understand one thing here simple thing: there are formulas for solving elementary trigonometric equations, just a short summary of the answers. For this brevity, we had to insert plus/minus into the cosine solution and (-1) n into the sine solution.
These inserts do not interfere in any way in tasks where you just need to write down the answer to an elementary equation. But if you need to solve an inequality, or then you need to do something with the answer: select roots on an interval, check for ODZ, etc., these insertions can easily unsettle a person.
So what should I do? Yes, either write the answer in two series, or solve the equation/inequality using the trigonometric circle. Then these insertions disappear and life becomes easier.)
We can summarize.
To solve the simplest trigonometric equations, there are ready-made answer formulas. Four pieces. They are good for instantly writing down the solution to an equation. For example, you need to solve the equations:
sinx = 0.3
Easily: x = (-1) n arcsin 0.3 + π n, n ∈ Z
cosx = 0.2
No problem: x = ± arccos 0.2 + 2π n, n ∈ Z
tgx = 1.2
Easily: x = arctan 1,2 + π n, n ∈ Z
ctgx = 3.7
One left: x= arcctg3,7 + π n, n ∈ Z
cos x = 1.8
If you, shining with knowledge, instantly write the answer:
x= ± arccos 1.8 + 2π n, n ∈ Z
then you are already shining, this... that... from a puddle.) Correct answer: there are no solutions. Don't understand why? Read what arc cosine is. In addition, if on the right side of the original equation there are tabular values of sine, cosine, tangent, cotangent, - 1; 0; √3; 1/2; √3/2 etc. - the answer through the arches will be unfinished. Arches must be converted to radians.
And if you come across inequality, like
then the answer is:
x πn, n ∈ Z
there is rare nonsense, yes...) Here you need to solve using the trigonometric circle. What we will do in the corresponding topic.
For those who heroically read to these lines. I simply cannot help but appreciate your titanic efforts. Bonus for you.)
Bonus:
When writing down formulas in an alarming combat situation, even seasoned nerds often get confused about where πn, and where 2π n. Here's a simple trick for you. In everyone formulas worth πn. Except for the only formula with arc cosine. It's standing there 2πn. Two peen. Keyword - two. In this same formula there are two sign at the beginning. Plus and minus. And there, and there - two.
So if you wrote two sign before the arc cosine, it’s easier to remember what will happen at the end two peen. And it also happens the other way around. The person will miss the sign ± , gets to the end, writes correctly two Pien, and he’ll come to his senses. There's something ahead two sign! The person will return to the beginning and correct the mistake! Like this.)
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
When solving many mathematical problems , especially those that occur before grade 10, the order of actions performed that will lead to the goal is clearly defined. Such problems include, for example, linear and quadratic equations, linear and quadratic inequalities, fractional equations and equations that reduce to quadratic. The principle of successfully solving each of the mentioned problems is as follows: you need to establish what type of problem you are solving, remember the necessary sequence of actions that will lead to the desired result, i.e. answer and follow these steps.
It is obvious that success or failure in solving a particular problem depends mainly on how correctly the type of equation being solved is determined, how correctly the sequence of all stages of its solution is reproduced. Of course, in this case it is necessary to have the skills to perform identical transformations and calculations.
The situation is different with trigonometric equations. It is not at all difficult to establish the fact that the equation is trigonometric. Difficulties arise when determining the sequence of actions that would lead to the correct answer.
By appearance equation, it is sometimes difficult to determine its type. And without knowing the type of equation, it is almost impossible to choose the right one from several dozen trigonometric formulas.
To solve a trigonometric equation, you need to try:
1. bring all functions included in the equation to “the same angles”;
2. bring the equation to “identical functions”;
3. factor the left side of the equation, etc.
Let's consider basic methods for solving trigonometric equations.
I. Reduction to the simplest trigonometric equations
Solution diagram
Step 1. Express a trigonometric function in terms of known components.
Step 2. Find the function argument using the formulas:
cos x = a; x = ±arccos a + 2πn, n ЄZ.
sin x = a; x = (-1) n arcsin a + πn, n Є Z.
tan x = a; x = arctan a + πn, n Є Z.
ctg x = a; x = arcctg a + πn, n Є Z.
Step 3. Find the unknown variable.
Example.
2 cos(3x – π/4) = -√2.
Solution.
1) cos(3x – π/4) = -√2/2.
2) 3x – π/4 = ±(π – π/4) + 2πn, n Є Z;
3x – π/4 = ±3π/4 + 2πn, n Є Z.
3) 3x = ±3π/4 + π/4 + 2πn, n Є Z;
x = ±3π/12 + π/12 + 2πn/3, n Є Z;
x = ±π/4 + π/12 + 2πn/3, n Є Z.
Answer: ±π/4 + π/12 + 2πn/3, n Є Z.
II. Variable replacement
Solution diagram
Step 1. Reduce the equation to algebraic form with respect to one of the trigonometric functions.
Step 2. Denote the resulting function by the variable t (if necessary, introduce restrictions on t).
Step 3. Write down and solve the resulting algebraic equation.
Step 4. Make a reverse replacement.
Step 5. Solve the simplest trigonometric equation.
Example.
2cos 2 (x/2) – 5sin (x/2) – 5 = 0.
Solution.
1) 2(1 – sin 2 (x/2)) – 5sin (x/2) – 5 = 0;
2sin 2 (x/2) + 5sin (x/2) + 3 = 0.
2) Let sin (x/2) = t, where |t| ≤ 1.
3) 2t 2 + 5t + 3 = 0;
t = 1 or e = -3/2, does not satisfy the condition |t| ≤ 1.
4) sin(x/2) = 1.
5) x/2 = π/2 + 2πn, n Є Z;
x = π + 4πn, n Є Z.
Answer: x = π + 4πn, n Є Z.
III. Equation order reduction method
Solution diagram
Step 1. Replace given equation linear, using the formulas for reducing the degree:
sin 2 x = 1/2 · (1 – cos 2x);
cos 2 x = 1/2 · (1 + cos 2x);
tg 2 x = (1 – cos 2x) / (1 + cos 2x).
Step 2. Solve the resulting equation using methods I and II.
Example.
cos 2x + cos 2 x = 5/4.
Solution.
1) cos 2x + 1/2 · (1 + cos 2x) = 5/4.
2) cos 2x + 1/2 + 1/2 · cos 2x = 5/4;
3/2 cos 2x = 3/4;
2x = ±π/3 + 2πn, n Є Z;
x = ±π/6 + πn, n Є Z.
Answer: x = ±π/6 + πn, n Є Z.
IV. Homogeneous equations
Solution diagram
Step 1. Reduce this equation to the form
a) a sin x + b cos x = 0 (homogeneous equation of the first degree)
or to the view
b) a sin 2 x + b sin x · cos x + c cos 2 x = 0 (homogeneous equation of the second degree).
Step 2. Divide both sides of the equation by
a) cos x ≠ 0;
b) cos 2 x ≠ 0;
and get the equation for tan x:
a) a tan x + b = 0;
b) a tan 2 x + b arctan x + c = 0.
Step 3. Solve the equation using known methods.
Example.
5sin 2 x + 3sin x cos x – 4 = 0.
Solution.
1) 5sin 2 x + 3sin x · cos x – 4(sin 2 x + cos 2 x) = 0;
5sin 2 x + 3sin x · cos x – 4sin² x – 4cos 2 x = 0;
sin 2 x + 3sin x · cos x – 4cos 2 x = 0/cos 2 x ≠ 0.
2) tg 2 x + 3tg x – 4 = 0.
3) Let tg x = t, then
t 2 + 3t – 4 = 0;
t = 1 or t = -4, which means
tg x = 1 or tg x = -4.
From the first equation x = π/4 + πn, n Є Z; from the second equation x = -arctg 4 + πk, k Є Z.
Answer: x = π/4 + πn, n Є Z; x = -arctg 4 + πk, k Є Z.
V. Method of transforming an equation using trigonometric formulas
Solution diagram
Step 1. Using all sorts of trigonometric formulas, reduce this equation to an equation solved by methods I, II, III, IV.
Step 2. Solve the resulting equation using known methods.
Example.
sin x + sin 2x + sin 3x = 0.
Solution.
1) (sin x + sin 3x) + sin 2x = 0;
2sin 2x cos x + sin 2x = 0.
2) sin 2x (2cos x + 1) = 0;
sin 2x = 0 or 2cos x + 1 = 0;
From the first equation 2x = π/2 + πn, n Є Z; from the second equation cos x = -1/2.
We have x = π/4 + πn/2, n Є Z; from the second equation x = ±(π – π/3) + 2πk, k Є Z.
As a result, x = π/4 + πn/2, n Є Z; x = ±2π/3 + 2πk, k Є Z.
Answer: x = π/4 + πn/2, n Є Z; x = ±2π/3 + 2πk, k Є Z.
The ability and skill to solve trigonometric equations is very 
important, their development requires significant effort, both on the part of the student and on the part of the teacher.
Many problems of stereometry, physics, etc. are associated with the solution of trigonometric equations. The process of solving such problems embodies many of the knowledge and skills that are acquired by studying the elements of trigonometry.
Trigonometric equations occupy an important place in the process of learning mathematics and personal development in general.
Still have questions? Don't know how to solve trigonometric equations?
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Reference information on the trigonometric functions sine (sin x) and cosine (cos x). Geometric definition, properties, graphs, formulas. Table of sines and cosines, derivatives, integrals, series expansions, secant, cosecant. Expressions through complex variables. Connection with hyperbolic functions.
Geometric definition of sine and cosine
|BD|- length of the arc of a circle with center at a point A.
α
- angle expressed in radians.
Definition
Sine (sin α) is a trigonometric function depending on the angle α between the hypotenuse and the leg right triangle, equal to the ratio length of the opposite side |BC| to the length of the hypotenuse |AC|.
Cosine (cos α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the hypotenuse |AC|.
Accepted notations
;
;
.
;
;
.
Graph of the sine function, y = sin x
Graph of the cosine function, y = cos x
Properties of sine and cosine
Periodicity
Functions y = sin x and y = cos x periodic with period 2π.
Parity
The sine function is odd. The cosine function is even.
Domain of definition and values, extrema, increase, decrease
The sine and cosine functions are continuous in their domain of definition, that is, for all x (see proof of continuity). Their main properties are presented in the table (n - integer).
| y= sin x | y= cos x | |
| Scope and continuity | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Range of values | -1 ≤ y ≤ 1 | -1 ≤ y ≤ 1 |
| Increasing | ||
| Descending | ||
| Maxima, y = 1 | ||
| Minima, y = - 1 | ||
| Zeros, y = 0 | ||
| Intercept points with the ordinate axis, x = 0 | y= 0 | y= 1 |
Basic formulas
Sum of squares of sine and cosine
Formulas for sine and cosine from sum and difference
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;
Formulas for the product of sines and cosines
Sum and difference formulas
Expressing sine through cosine
;
;
;
.
Expressing cosine through sine
;
;
;
.
Expression through tangent
; .
When , we have:
;
.
At :
;
.
Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for certain values of the argument. 
Expressions through complex variables
;
Euler's formula
{ -∞ < x < +∞ }
Secant, cosecant
Inverse functions
Inverse functions to sine and cosine are arcsine and arccosine, respectively.
Arcsine, arcsin
Arccosine, arccos
Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.