How to Determine the Value of x Using a Trigonometric Ratio

Oct 13, 2025 by talynt.org 148 views

Introduction

Trigonometric ratios are essential in mathematics, particularly in solving problems involving right-angled triangles. These ratios help us find the length of sides and angles in a triangle. In this article, we will explore how to determine the value of x using a trigonometric ratio.

What are Trigonometric Ratios?

Trigonometric ratios are the relationships between the angles and the sides of a right-angled triangle. There are six basic trigonometric ratios: sine, cosine, tangent, cotangent, secant, and cosecant. These ratios are defined as follows:

Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

How to Determine the Value of x Using a Trigonometric Ratio

To determine the value of x using a trigonometric ratio, we need to follow these steps:

Identify the given information: We need to identify the given information, which includes the trigonometric ratio, the angle, and the side length.
Choose the correct trigonometric ratio: We need to choose the correct trigonometric ratio based on the given information.
Set up the equation: We need to set up the equation using the trigonometric ratio and the given information.
Solve for x: We need to solve for x using the equation.

Example 1: Using the Sine Ratio

Suppose we have a right-angled triangle with an angle of 30° and a side length of 10 cm. We want to find the length of the side opposite the angle.

Step 1: Identify the given information

Trigonometric ratio: sine
Angle: 30°
Side length: 10 cm

Step 2: Choose the correct trigonometric ratio

Since we are given the angle and the side length, we can use the sine ratio.

Step 3: Set up the equation

sin(30°) = opposite side / hypotenuse
sin(30°) = x / 10

Step 4: Solve for x

x = 10 * sin(30°)
x = 10 * 0.5
x = 5 cm

Example 2: Using the Cosine Ratio

Suppose we have a right-angled triangle with an angle of 45° and a side length of 12 cm. We want to find the length of the side adjacent to the angle.

Step 1: Identify the given information

Trigonometric ratio: cosine
Angle: 45°
Side length: 12 cm

Step 2: Choose the correct trigonometric ratio

Since we are given the angle and the side length, we can use the cosine ratio.

Step 3: Set up the equation

cos(45°) = adjacent side / hypotenuse
cos(45°) = x / 12

Step 4: Solve for x

x = 12 * cos(45°)
x = 12 * 0.7071
x = 8.485 cm

Conclusion

In conclusion, trigonometric ratios are essential in mathematics, particularly in solving problems involving right-angled triangles. By following the steps outlined in this article, we can determine the value of x using a trigonometric ratio. We can use the sine, cosine, and tangent ratios to find the length of sides and angles in a triangle.

Answer

Based on the examples provided, the correct answer is:

A) 5 cm (Example 1)
C) 8.49 cm (Example 2)

Introduction

In our previous article, we explored how to determine the value of x using a trigonometric ratio. In this article, we will answer some frequently asked questions related to trigonometric ratios and provide additional examples to help you understand the concept better.

Q: What is the difference between the sine, cosine, and tangent ratios?

A: The sine, cosine, and tangent ratios are all trigonometric ratios that help us find the length of sides and angles in a right-angled triangle. The main difference between them is the side length they use:

Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: How do I choose the correct trigonometric ratio?

A: To choose the correct trigonometric ratio, you need to identify the given information and the side length you want to find. Then, you can choose the ratio that uses the given side length.

Q: What is the range of values for the sine, cosine, and tangent ratios?

A: The range of values for the sine, cosine, and tangent ratios is:

Sine (sin): -1 ≤ sin(x) ≤ 1
Cosine (cos): -1 ≤ cos(x) ≤ 1
Tangent (tan): -∞ < tan(x) < ∞

Q: How do I solve for x using a trigonometric ratio?

A: To solve for x using a trigonometric ratio, you need to follow these steps:

Identify the given information: Identify the given information, which includes the trigonometric ratio, the angle, and the side length.
Choose the correct trigonometric ratio: Choose the correct trigonometric ratio based on the given information.
Set up the equation: Set up the equation using the trigonometric ratio and the given information.
Solve for x: Solve for x using the equation.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

Pythagorean identity: sin^2(x) + cos^2(x) = 1
Quotient identity: tan(x) = sin(x) / cos(x)
Reciprocal identity: csc(x) = 1 / sin(x), sec(x) = 1 / cos(x), cot(x) = 1 / tan(x)

Q: How do I use trigonometric ratios to solve problems involving right-angled triangles?

A: To use trigonometric ratios to solve problems involving right-angled triangles, you need to follow these steps:

Identify the given information: Identify the given information, which includes the trigonometric ratio, the angle, and the side length.
Choose the correct trigonometric ratio: Choose the correct trigonometric ratio based on the given information.
Set up the equation: Set up the equation using the trigonometric ratio and the given information.
Solve for x: Solve for x using the equation.

Conclusion

In conclusion, trigonometric ratios are essential in mathematics, particularly in solving problems involving right-angled triangles. By understanding the sine, cosine, and tangent ratios, you can solve a wide range of problems involving right-angled triangles. We hope this Q&A article has helped you understand the concept better and provided you with the tools you need to solve problems involving trigonometric ratios.

Additional Examples

Here are some additional examples to help you understand the concept better:

Example 1: Find the length of the side opposite the angle in a right-angled triangle with an angle of 60° and a side length of 15 cm.
Example 2: Find the length of the side adjacent to the angle in a right-angled triangle with an angle of 45° and a side length of 20 cm.
Example 3: Find the length of the side opposite the angle in a right-angled triangle with an angle of 30° and a side length of 10 cm.

Answer

Based on the examples provided, the correct answers are:

Example 1: 12.5 cm
Example 2: 14.14 cm
Example 3: 5 cm