Problem number 260 from a geometry textbook for the ninth grade, edited by A.G. Merzlyak requires the student to accurately apply the law of cosines to find the unknown side of a triangle when the other two sides and the angle between them are known. The condition typically gives specific numerical values, such as sides $a$ and $b$, and angle $\gamma$, and requires the length of side $c$ to be calculated with a high degree of accuracy, avoiding common arithmetic errors when working with trigonometric functions.
A misunderstanding of which theorem should be used in a given data configuration often leads to attempts to apply the Pythagorean theorem, which is categorically false for arbitrary triangles that are not right triangles. The correct algorithm of actions involves writing the basic formula $c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma)$, substituting the initial data from the condition and sequentially performing calculations, preserving the signs and order of operations.
Reviewing this issue is important not only for obtaining a grade, but also for developing a stable skill in working with trigonometric relationships, which will be critically needed when studying vectors and solving stereometric problems in high school. Attention It is worth paying attention to the units of measurement of angles (degrees or radians) and the correct rounding of the final answer according to the teacher’s requirements.
Analysis of the condition and choice of solution method
The first step in working on exercise 260 is to carefully analyze the geometric figure presented in the condition. Usually this is an arbitrary triangle $ABC$, where the lengths of the sides $AB$ and $AC$ (or $BC$) and the angle at the vertex $A$ (or another vertex) are given. Cosine theorem is a universal tool in such cases, allowing you to tie three sides and one corner.
It is necessary to clearly identify which angle is opposite to the desired side. If we are looking for side $c$, then we need the angle $\gamma$ (gamma) lying opposite it. Confusion in the designations of sides and angles is a common reason for obtaining incorrect results, so it is recommended to immediately make a schematic drawing.
- 📐 Determine known quantities: write down the lengths of two sides and the degree measure of the angle between them.
- 📐 Find out which side of the triangle you need to find in the answer to the question of the problem.
- 📐 Check whether the values of trigonometric functions are given in the textbook reference materials or you need to calculate them yourself.
It is important to understand that the cosine theorem is a generalization of the Pythagorean theorem. If the angle between the sides is 90 degrees, the cosine of that angle is zero, and the formula becomes the classic equality of squares. However, in problem 260 the angle is, as a rule, acute or obtuse, which requires that the term with the cosine be taken into account.
Step-by-step calculation algorithm
The process of solving problem No. 260 requires a strict sequence of actions. First you need to square the known lengths of the sides. Then the product of twice the product of these sides and the cosine of the known angle is calculated. The key point here is the correct substitution of the cosine sign, especially if the angle is obtuse (more than 90 degrees), since the cosine of an obtuse angle is a negative value.
After performing all the arithmetic operations on the right side of the equation, the square of the desired side is obtained. To find the length itself, you need to take the square root of the resulting number. Rounding errors often occur at this stage, so it is better to perform intermediate calculations with a reserve of decimal places.
☑️ Checklist for solving a problem
If in the problem statement the values of the sides are given in the form of roots or fractions, it is advisable to carry out calculations in symbolic form until the very end in order to avoid loss of accuracy. Only in the final line of the answer can you switch to decimal fractions if the recording format requires it.
⚠️ Attention: When calculating the cosine value of an angle, make sure that your calculator or Bradis table is set to work in degrees, not radians, if the angle is given in degrees.
Application of the cosine theorem in practice
Let's consider a specific example typical of problem 260. Let a triangle $ABC$ be given, where side $a = 5$ cm, side $b = 7$ cm, and angle $\gamma = 60^\circ$. We need to find side $c$. The formula takes the form: $c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(60^\circ)$.
We perform the calculations: $25 + 49 - 70 \cdot 0.5$. Knowing that $\cos(60^\circ) = 0.5$, we get $74 - 35 = 39$. Therefore, $c^2 = 39$, and the required side $c = \sqrt{39}$. In this case, an exact answer will be preferable to an approximate one, although a decimal fraction will be required to construct the drawing.
In more complex versions of the problem, the angle may not be tabulated, and then the cosine value will have to be taken from tables or calculated using an engineering calculator. A.G. Merzlyak often includes problems in the collection where the answer is an integer or a “beautiful” root, which serves as a good check of the correctness of the calculations.
Table values of cosines
cos(30°) = √3/2 ≈ 0.866; cos(45°) = √2/2 ≈ 0.707; cos(60°) = 1/2 = 0.5; cos(90°) = 0.
The practical application of this theorem goes far beyond the school curriculum. Engineers use similar calculations when designing bridge trusses, where it is important to know the lengths of the elements at given angles of inclination. Understanding the physics of the process helps to better remember the mathematical abstraction.
Typical mistakes students make when solving
An analysis of the work of ninth-graders shows that the most common mistake when solving problem 260 is the sign before the last term in the formula. Students often forget the minus and write a plus, which radically changes the result. It is necessary to remember: in the cosine theorem there is always a minus sign, unlike the Pythagorean theorem.
The second common mistake is confusion between sine and cosine. The sine theorem applies to other sets of data (for example, when given two angles and a side), and using it here will result in an incorrect answer. Discriminant This has nothing to do with quadratic equations, but confusion of formulas happens regularly.
- ❌ They forget to multiply the product of the sides by 2.
- ❌ The angle opposite to the desired side is incorrectly determined.
- ❌ Intermediate results are rounded, which introduces a large error into the final.
It is also worth mentioning the error in units of measurement. If the sides are given in centimeters and the angle in radians (although this is rare in 9th grade), or if one angle is given in degrees and the other in minutes, it is necessary to bring all the data to a single standard before starting the calculations.
Comparison of triangle solving methods
To gain a deep understanding of the topic, it is useful to compare the cosine theorem with other methods. Below is a table demonstrating when which method is applicable in the context of a 9th grade geometry course.
| Method | Given | Find | Formula |
|---|---|---|---|
| Cosine theorem | Two sides and the angle between them | Third party | $c^2 = a^2 + b^2 - 2ab \cos \gamma$ |
| Theorem of sines | Side and two corners | Other parties | $a / \sin \alpha = b / \sin \beta$ |
| Pythagorean theorem | Two legs (rectangular triangular) | Hypotenuse | $c^2 = a^2 + b^2$ |
| Sum of angles | Two angles of a triangle | Third corner | $\alpha + \beta + \gamma = 180^\circ$ |
Using a table helps to systematize knowledge and quickly navigate the conditions of the task. Problem 260 belongs to the first type, where you cannot do without the cosine theorem if the triangle is not right-angled. In a right triangle, the cosine theorem also works, but turns into the Pythagorean theorem, since $\cos(90^\circ) = 0$.
Understanding the relationship between these methods allows you to solve more complex combined problems where you need to find the area of a triangle, the radius of a circumscribed circle, or the length of the median using the found sides. The main thing is not to try to apply the method of sines when given two sides and the angle between them, as this will lead to an equation with two unknowns.
Checking the result and logical control
After obtaining a numerical answer, a feasibility check must be performed. The side of a triangle cannot be negative or equal to zero. The triangle inequality also works: the sum of any two sides is always greater than the third side. If your answer violates this rule, then there is a computational error somewhere.
Logical control also includes estimating the angle value. If the angle is obtuse, the opposite side should be the largest in the triangle. If the angle is acute, the side can be any, but in Problem 260, where the angle is usually given between known sides, this rule helps check the order of magnitude.
Tip: Always make a rough hand drawing before deciding. This will help you visually assess the proportions and notice gross errors in calculations.
Another way to check is to substitute the result back into the formula, or use an alternative method if possible (for example, omitting the height and calculating through right triangles, although this is much longer). In a test environment, a quick visual inspection of the drawing is your best friend.
⚠️ Attention: If, when extracting the square root, you get a negative number under the root, it means that you made an error in signs or arithmetic, since the length of a side is a real and positive quantity.
Significance of the task for further learning
The skills practiced in Problem No. 260 are fundamental to the physics course that begins at the same time or a little later. Finding the resultant force, decomposing the velocity vectors into components - all this is based on the same trigonometric dependencies. Vector algebra in 11th grade it also relies on the scalar product, the formula of which is structurally identical to the cosine theorem.
The ability to work with formulas containing squares and roots develops mathematical culture and attentiveness. Errors in signs or procedures made now can come back to haunt you when solving logarithmic equations or derivatives in high school. Therefore, it is worth approaching the design of a solution to Problem 260 with the utmost seriousness.
For those who plan to take the OGE or the Unified State Exam in mathematics, confident knowledge of trigonometry in a triangle is a mandatory minimum. Problems in finished drawings often require instant recognition of the “two sides and an angle between them” situation, which is tested in issue 260.
Conclusion: Problem 260 is a classic simulator for applying the cosine theorem, which forms the basis for solving complex geometric and physical problems.
Do I need to learn the cosine theorem by heart?
Yes, the formula $c^2 = a^2 + b^2 - 2ab \cos \gamma$ must be known by heart, since it is reference material that must be brought to automaticity. Unlike complex integrals, this formula is used constantly.
Is it possible to solve the 260 problem using height?
Theoretically it is possible by dropping the height from one vertex and getting two right triangles. However, this will significantly complicate the calculations and take more time. The cosine theorem provides a direct and fast path to the answer.
What to do if the cosine of the angle is negative?
If the angle is obtuse (more than 90 degrees), its cosine is negative. In the formula there is a minus before the product, so minus times minus will give a plus. This means that the square of the third side will be greater than the sum of the squares of the other two, which is logical for an obtuse triangle.
Where can I find tables of values of trigonometric functions?
Bradis tables or basic values of trigonometric functions are usually given at the end of a geometry textbook Merzlyak or in school reference books. For standard angles (30, 45, 60, 90), it is better to learn the values.