Solving Trigonometric And Geometry Problems A Step-by-Step Guide

This article delves into the solutions of two distinct mathematical problems, offering a detailed, step-by-step approach to ensure clarity and understanding. We will tackle a trigonometric expression and a geometry problem, showcasing the fundamental principles and techniques involved in solving them. Whether you're a student looking to enhance your problem-solving skills or simply someone with a keen interest in mathematics, this guide will provide valuable insights and practical knowledge.

a) Evaluating the Trigonometric Expression: $-\frac{4}{3} + \sin 30^{\circ} + \operatorname{tg} 45^{\circ}$

In this section, we will break down the evaluation of the trigonometric expression $-\frac{4}{3} + \sin 30^{\circ} + \operatorname{tg} 45^{\circ}$. This problem combines basic arithmetic operations with trigonometric functions, requiring a solid understanding of both. Our approach will be methodical, first recalling the standard values of trigonometric functions for specific angles and then performing the arithmetic operations. By the end of this section, you will have a clear understanding of how to solve similar expressions.

Step 1: Understanding the Trigonometric Functions

Before diving into the calculation, it's crucial to understand the trigonometric functions involved. Specifically, we need to know the values of $\sin 30^{\circ}$ and $\operatorname{tg} 45^{\circ}$. These are standard values that are often memorized or can be derived from the unit circle or special right triangles. Let's briefly review these functions:

• Sine (sin): In a right-angled triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Tangent (tg or tan): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Step 2: Recalling Standard Trigonometric Values

Now, let's recall the values of $\sin 30^{\circ}$ and $\operatorname{tg} 45^{\circ}$. These are fundamental values in trigonometry:

• $\sin 30^{\circ} = \frac{1}{2}$
• $\operatorname{tg} 45^{\circ} = 1$

These values can be derived from the 30-60-90 and 45-45-90 special right triangles, respectively. For $\sin 30^{\circ}$, consider a 30-60-90 triangle. The side opposite the 30-degree angle is half the length of the hypotenuse, hence the value of $\frac{1}{2}$. For $\operatorname{tg} 45^{\circ}$, in a 45-45-90 triangle, the opposite and adjacent sides are equal, making the tangent 1.

Step 3: Substituting the Values into the Expression

With the trigonometric values in hand, we can substitute them into the original expression:

$-\frac{4}{3} + \sin 30^{\circ} + \operatorname{tg} 45^{\circ} = -\frac{4}{3} + \frac{1}{2} + 1$

This substitution transforms the expression into a simple arithmetic problem involving fractions and integers. The next step is to find a common denominator to combine these terms.

Step 4: Finding a Common Denominator

To add or subtract fractions, they must have a common denominator. In this case, the denominators are 3 and 2. The least common multiple (LCM) of 3 and 2 is 6. Therefore, we will convert each fraction to have a denominator of 6:

• $-\frac{4}{3} = -\frac{4 \times 2}{3 \times 2} = -\frac{8}{6}$
• $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$
• $1 = \frac{6}{6}$

Now the expression looks like this:

$-\frac{8}{6} + \frac{3}{6} + \frac{6}{6}$

Step 5: Performing the Arithmetic Operations

Now that all terms have a common denominator, we can perform the addition and subtraction:

$-\frac{8}{6} + \frac{3}{6} + \frac{6}{6} = \frac{-8 + 3 + 6}{6} = \frac{1}{6}$

Therefore, the value of the expression $-\frac{4}{3} + \sin 30^{\circ} + \operatorname{tg} 45^{\circ}$ is $\frac{1}{6}$. This completes the evaluation of the trigonometric expression.

Summary of the Solution

To summarize, we evaluated the expression $-\frac{4}{3} + \sin 30^{\circ} + \operatorname{tg} 45^{\circ}$ by:

1. Recalling the standard trigonometric values: $\sin 30^{\circ} = \frac{1}{2}$ and $\operatorname{tg} 45^{\circ} = 1$.
2. Substituting these values into the expression.
3. Finding a common denominator to combine the fractions.
4. Performing the arithmetic operations to arrive at the final answer, $\frac{1}{6}$.

This step-by-step approach highlights the importance of understanding fundamental trigonometric values and the rules of arithmetic operations when solving mathematical problems. In the next section, we will tackle a geometry problem involving the diagonal of a square.

b) Finding the Diagonal of a Square

This section focuses on determining the length of the diagonal of a square with a side length of 7 cm. This problem is a classic example of applying the Pythagorean theorem in geometry. We will walk through the steps, explaining the theorem and its application in this specific context. Understanding this concept is crucial for solving various geometric problems involving squares, rectangles, and other related shapes.

Step 1: Understanding the Properties of a Square

Before we calculate the diagonal, let's review the properties of a square. A square is a quadrilateral with four equal sides and four right angles (90 degrees). This means that all sides have the same length, and the angles at each corner are right angles. The diagonal of a square is a line segment that connects two non-adjacent vertices (corners). Importantly, the diagonal divides the square into two right-angled triangles.

Step 2: Applying the Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that relates the sides of a right-angled triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as:

$a^2 + b^2 = c^2$

where:

• a and b are the lengths of the two shorter sides (legs) of the right-angled triangle.
• c is the length of the hypotenuse.

In our case, the diagonal of the square acts as the hypotenuse of the right-angled triangles formed by the diagonal. The sides of the square are the legs of these triangles. Therefore, we can apply the Pythagorean theorem to find the length of the diagonal.

Step 3: Setting up the Equation

Let's denote the side length of the square as s, which is given as 7 cm. Let's denote the length of the diagonal as d. Since the diagonal divides the square into two right-angled triangles, and the sides of the square are the legs of these triangles, we can set up the equation using the Pythagorean theorem:

$s^2 + s^2 = d^2$

Substituting the given side length (s = 7 cm) into the equation, we get:

$7^2 + 7^2 = d^2$

Step 4: Solving for the Diagonal

Now, let's solve the equation for d:

$49 + 49 = d^2$

$98 = d^2$

To find d, we take the square root of both sides:

$d = \sqrt{98}$

We can simplify the square root by factoring 98:

$98 = 2 \times 49 = 2 \times 7^2$

So,

$d = \sqrt{2 \times 7^2} = 7\sqrt{2}$

Therefore, the length of the diagonal of the square is $7\sqrt{2}$ cm.

Step 5: Approximating the Value (Optional)

While $7\sqrt{2}$ cm is the exact length of the diagonal, it can be helpful to approximate this value for practical purposes. The square root of 2 is approximately 1.414. So,

$d \approx 7 \times 1.414 \approx 9.898$

Therefore, the diagonal is approximately 9.898 cm. However, we will stick to the exact value for our final answer.

Summary of the Solution

To summarize, we found the length of the diagonal of a square with a side length of 7 cm by:

1. Understanding the properties of a square and recognizing that the diagonal divides it into two right-angled triangles.
2. Applying the Pythagorean theorem: $a^2 + b^2 = c^2$.
3. Setting up the equation: $7^2 + 7^2 = d^2$.
4. Solving for the diagonal: $d = \sqrt{98} = 7\sqrt{2}$ cm.

This detailed explanation demonstrates how the Pythagorean theorem can be applied to solve geometric problems involving squares and other right-angled shapes. The exact length of the diagonal is $7\sqrt{2}$ cm.

Conclusion

In this article, we tackled two distinct mathematical problems: a trigonometric expression and a geometry problem. We systematically broke down each problem, explaining the concepts and steps involved in finding the solutions. For the trigonometric expression $-\frac{4}{3} + \sin 30^{\circ} + \operatorname{tg} 45^{\circ}$, we found the value to be $\frac{1}{6}$. For the geometry problem, we determined that the diagonal of a square with a side length of 7 cm is $7\sqrt{2}$ cm. These examples highlight the importance of understanding fundamental mathematical principles and applying them methodically to solve problems.

By understanding the properties of squares and the Pythagorean theorem, we were able to accurately calculate the diagonal's length. This underscores the importance of a solid grasp of geometric principles in problem-solving.

Through these detailed solutions, we hope to have provided a valuable resource for students and anyone interested in enhancing their mathematical skills. By following a step-by-step approach and understanding the underlying concepts, complex problems can be made more manageable and solvable. Continue practicing and exploring different types of mathematical problems to further develop your proficiency and confidence.