Basic theoretical information
Some recommendations for performing trigonometric transformations
When performing trigonometric transformations, follow these tips:
- Do not immediately try to come up with an example solution from the beginning to the end.
- Do not attempt to convert the whole example at once. Move forward in small steps.
- Remember that in addition to trigonometric formulas in trigonometry, you can still apply all fair algebraic transformations (putting out the bracket, reducing fractions, abbreviated multiplication formulas, etc.).
- Believe that everything will be fine.
Basic Trigonometric Formulas
Most formulas in trigonometry are often used both from right to left and from left to right, so you need to learn these formulas so well that you can easily apply a certain formula in both directions. We write to begin the definition of trigonometric functions. Let there be a right triangle:
Then, the definition of sine:
Determination of cotangent:
Basic trigonometric identity:
The simplest corollaries of the basic trigonometric identity are:
Double Angle Formulas. Dual Angle Sine:
Cosine of double angle:
Dual Angle Tangent:
Cotangent double angle:
Additional trigonometric formulas
Trigonometric addition formulas. Sine Amount:
Trigonometric formulas for converting a sum into a product. Sum of Sines:
Amount of tangents:
Amount of cotangents:
Trigonometric formulas for converting a work to a sum. The work of sines:
The product of sine and cosine:
The product of cosines:
Degree reduction formulas. Degree Formula for Sinus:
Degree formula for cosine:
The formula for reducing the degree of tangent:
Decrease formula for cotangents:
Half angle formulas. Half-angle formula for tangent:
Half Angle Formula for Cotangent:
Trigonometric Reduction Formulas
The cosine function is called the sine function and vice versa. Similarly, the tangent and cotangent functions are co-functions. Formulas of reduction can be formulated as the following rule:
- If in the reduction formula the angle is subtracted (added) from 90 degrees or 270 degrees, then the reduced function changes to a function;
- If in the reduction formula the angle is subtracted (added) from 180 degrees or 360 degrees, then the name of the function being retained is preserved;
- At the same time, the preceding function is preceded by that sign, which has a reducible (i.e., initial) function in the corresponding quarter, if we consider the subtracted (added) angle as sharp.
Reduction formulas are given in the form of a table:
For trigonometric circles, it is easy to determine the tabular values of trigonometric functions:
To solve a trigonometric equation, it must be reduced to one of the simplest trigonometric equations, which will be discussed below. For this:
- You can apply the trigonometric formulas above. You do not need to try to convert the whole example at once, but you need to move forward in small steps.
- One should not forget about the possibility of transforming an expression with the help of algebraic methods, i.e. for example, to put something out of the bracket, or, conversely, open the parentheses, reduce the fraction, apply the reduced multiplication formula , reduce the fraction to a common denominator, and so on.
- When solving trigonometric equations, the grouping method can be applied . It should be remembered that in order for a product of several factors to be equal to zero, it suffices that each of them be equal to zero, and the others exist .
- Using the method of replacing a variable , as usual, the equation after introducing the replacement should become easier and not contain the original variable. You also need to remember to perform a reverse replacement.
- Remember that homogeneous equations are often found in trigonometry.
- When opening modules or solving irrational equations with trigonometric functions, it is necessary to remember and take into account all the subtleties of solving the corresponding equations with ordinary functions.
- Remember about the LDL (in the trigonometric equations, the restrictions on the DHS basically boil down to the fact that it is impossible to divide by zero, but do not forget about other constraints, especially about the positiveness of expressions in rational degrees and under the roots of even degrees). Also remember that sine and cosine values can only be between minus one and plus units inclusive.
The main thing, if you do not know what to do, do at least something, while the main thing is to correctly use trigonometric formulas. If what you get at the same time gets better and better, then continue the decision, and if it gets worse, then go back to the beginning and try to apply other formulas, so do until you stumble on the correct course of the decision.
Formulas for solving the simplest trigonometric equations. For sine, there are two equivalent solution recording forms:
For the remaining trigonometric functions, the entry is unambiguous. For cosine:
Solving trigonometric equations in some special cases: