Question

Помогите с решением!Желательно объяснить само решение.Тема Тригонометрические уравнения. Почему arccos 1/2= пи/3 -Объясните arccos 1 =0 Почему?? acos(-корень(2)/2) =3*пи/4 Объясните решение Может есть какая-то формула, по которой можно решать??

Answer 1

Trigonometric Equations

Trigonometric equations involve trigonometric functions such as sine, cosine, tangent, etc. Solving these equations often requires finding the values of angles that satisfy the given equation.

Explanation of arccos(1/2) = π/3

The arccosine function, denoted as arccos(x) or acos(x), is the inverse of the cosine function. It returns the angle whose cosine is equal to the given value.

In this case, we are looking for the angle whose cosine is equal to 1/2. The cosine function represents the ratio of the adjacent side to the hypotenuse in a right triangle. When the cosine is 1/2, it means that the adjacent side is half the length of the hypotenuse.

To find the angle whose cosine is 1/2, we can use the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The cosine of an angle is equal to the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

In the unit circle, the angle whose cosine is 1/2 is π/3 (or 60 degrees). This can be verified by calculating the cosine of π/3, which indeed equals 1/2.

Therefore, arccos(1/2) = π/3.

Explanation of arccos(1) = 0

The arccosine function returns the angle whose cosine is equal to the given value. The cosine function has a range of -1 to 1. When the cosine is 1, it means that the angle is 0 degrees or 2π radians.

Therefore, arccos(1) = 0.

Explanation of acos(-√2/2) = 3π/4

To find the angle whose cosine is equal to -√2/2, we can again use the unit circle. The cosine of an angle is equal to the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

In this case, the cosine is -√2/2, which means that the x-coordinate of the point is -√2/2. By examining the unit circle, we can see that this corresponds to an angle of 3π/4 (or 135 degrees).

Therefore, acos(-√2/2) = 3π/4.

General Approach to Solving Trigonometric Equations

There is no single formula that can be used to solve all trigonometric equations. The approach to solving these equations depends on the specific equation and the trigonometric function involved.

However, there are some common techniques that can be used:

1. Simplify the equation: Try to simplify the equation by applying trigonometric identities or algebraic manipulations. 2. Use inverse trigonometric functions: If the equation involves a trigonometric function, try applying the inverse trigonometric function to both sides of the equation to isolate the variable. 3. Solve for the variable: Once you have isolated the variable, solve for it using algebraic techniques or by using a calculator or trigonometric tables.

It's important to note that trigonometric equations can have multiple solutions, so it's necessary to check for all possible solutions within the given range.

I hope this explanation helps! Let me know if you have any further questions.