Trigonometric equations. As part of the math exam in the first part, there is a task related to solving an equation - these are simple equations that can be solved in minutes, many types can be solved orally. Includes: linear, quadratic, rational, irrational, exponential, logarithmic and trigonometric equations.
In this article, we will look at trigonometric equations. Their solution differs both in the amount of calculation and in complexity from the rest of the problems in this part. Don't be alarmed, the word "difficulty" refers to their relative difficulty compared to other tasks.
In addition to finding the roots of the equation themselves, it is necessary to determine the largest negative or smallest positive root. The probability that you will get a trigonometric equation on the exam is, of course, small.
They are less than 7% in this part of the exam. But that doesn't mean they should be ignored. In part C, it is also necessary to solve the trigonometric equation, so it is simply necessary to understand the solution method well and understand the theory.
Understanding the section "Trigonometry" in mathematics largely determines your success in solving many problems. I remind you that the answer is an integer or a final decimal. After you get the roots of the equation, ALWAYS do a check. It will not take much time, and you will save yourself from mistakes.
In the future, we will also look at other equations, don't miss it! Recall the formulas for the roots of trigonometric equations, you need to know them:
Knowing these values is necessary, this is the “alphabet”, without which it will be impossible to cope with many tasks. Great, if the memory is good, you easily learned and remembered these values. What to do if this does not work out, there is confusion in your head, but it’s just that you went astray during the exam. It will be a shame to lose a point due to the fact that you write down the wrong value in the calculations.
This value is simple, it is also given in the theory you received in the second letter after subscribing to the newsletter. If you haven't signed up yet, do it! In the future, we will also consider how these values can be determined from a trigonometric circle. It is not for nothing that it is called the “Golden Heart of Trigonometry”.
I will immediately explain, in order to avoid confusion, that in the equations considered below, definitions of the arcsine, arccosine, arctangent are given using the angle X for the corresponding equations: cosx=a, sinx=a, tgx=a, where X can also be an expression. In the examples below, we have the argument specified by the expression.
So, consider the following tasks:
Find the root of the equation:
Write down the largest negative root in your answer.
The solution to the equation cos x = a are two roots:
Definition: Let the number a modulo not exceed one. The arc cosine of the number a is the angle x, lying in the range from 0 to Pi, the cosine of which is equal to a.
Means
Express x:
Find the largest negative root. How to do it? We substitute different values of n into the obtained roots, calculate and choose the largest negative one.
We calculate:
With n \u003d - 2 x 1 \u003d 3 (- 2) - 4.5 \u003d - 10.5 x 2 \u003d 3 (- 2) - 5.5 \u003d - 11.5
With n \u003d - 1 x 1 \u003d 3 (- 1) - 4.5 \u003d - 7.5 x 2 \u003d 3 (- 1) - 5.5 \u003d - 8.5
At n = 0 x 1 = 3∙0 - 4.5 = - 4.5 x 2 = 3∙0 - 5.5 = - 5.5
At n \u003d 1 x 1 \u003d 3 1 - 4.5 \u003d - 1.5 x 2 \u003d 3 1 - 5.5 \u003d - 2.5
At n = 2 x 1 = 3∙2 - 4.5 = 1.5 x 2 = 3∙2 - 5.5 = 0.5
We found that the largest negative root is -1.5
Answer: -1.5
Decide for yourself:
Solve the equation:
The solution of the equation sin x = a are two roots:
Either (it combines both of the above):
Definition: Let the number a modulo not exceed one. The arcsine of the number a is the angle x, lying in the range from - 90 o to 90 o, the sine of which is equal to a.
Means
Express x (multiply both sides of the equation by 4 and divide by pi):
Find the smallest positive root. Here it is immediately clear that when substituting negative values of n, we will get negative roots. Therefore, we will substitute n = 0,1,2 ...
For n = 0 x = (- 1) 0 + 4∙0 + 3 = 4
For n = 1 x = (- 1) 1 + 4∙1 + 3 = 6
For n = 2 x = (- 1) 2 + 4∙2 + 3 = 12
Check for n = –1 x = (–1) –1 + 4∙(–1) + 3 = –2
So the smallest positive root is 4.
Answer: 4
Decide for yourself:
Solve the equation:
Write the smallest positive root for your answer.
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