Calculate The Length Of AC, Sin A, Cos A, And Tan C In A Right Triangle ABC Where AB = 8 And BC = 6.

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In mathematics, understanding the relationships between the sides and angles of a right triangle is fundamental. This article will walk you through the process of calculating the length of a side and various trigonometric ratios – sine, cosine, and tangent – without using a calculator. We will focus on a specific example: a right triangle ABC where AB = 8 and BC = 6. By the end of this guide, you’ll have a clear understanding of how to apply the Pythagorean theorem and trigonometric definitions to solve such problems.

1. Determining the Length of AC Using the Pythagorean Theorem

To determine the length of the hypotenuse AC, we will use the Pythagorean theorem. The Pythagorean theorem is a fundamental concept in Euclidean geometry that describes the relationship between the sides of a right triangle. It states that 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. In mathematical terms, this can be expressed as:

a² + b² = c²

Where:

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

In our given right triangle ABC, AB and BC are the two shorter sides, and AC is the hypotenuse. We are given that AB = 8 and BC = 6. To find the length of AC, we substitute these values into the Pythagorean theorem:

AC² = AB² + BC² AC² = 8² + 6² AC² = 64 + 36 AC² = 100

To find the length of AC, we take the square root of both sides of the equation:

AC = √100 AC = 10

Therefore, the length of AC is 10 units. This calculation demonstrates a direct application of the Pythagorean theorem, allowing us to find the length of the hypotenuse when the lengths of the other two sides are known. The Pythagorean theorem is crucial in various fields, including engineering, physics, and computer graphics, making its understanding essential for anyone studying these disciplines. Additionally, this simple yet powerful theorem serves as a foundation for more advanced concepts in trigonometry and geometry. By mastering this theorem, you can solve a wide range of problems involving right triangles and their side lengths.

2. Calculating Sin A

In trigonometry, the sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. To find sin A in the right triangle ABC, we need to identify the side opposite angle A and the hypotenuse. In this case, the side opposite angle A is BC, and the hypotenuse is AC. We are given that BC = 6, and we have already calculated that AC = 10 using the Pythagorean theorem.

The formula for sine is:

sin(angle) = Opposite / Hypotenuse

Applying this to angle A in triangle ABC:

sin A = BC / AC

Now, substitute the given values:

sin A = 6 / 10

Simplify the fraction:

sin A = 3 / 5

Thus, the sine of angle A is 3/5. This value represents the ratio of the side opposite angle A to the hypotenuse. The sine function is one of the fundamental trigonometric functions, playing a crucial role in various mathematical and scientific applications. It helps us understand the relationship between angles and sides in triangles, which is vital in fields such as navigation, physics, and engineering. For example, in physics, the sine function is used to calculate the components of vectors, which are essential in understanding forces and motion. In engineering, it is used in structural analysis and design. Furthermore, the concept of sine is not limited to right triangles; it extends to more complex trigonometric functions and is used extensively in calculus and other advanced mathematical topics. Understanding how to calculate sin A is therefore a fundamental skill in mathematics and its applications.

3. Determining Cos A

Cos A, or the cosine of angle A, is another essential trigonometric ratio that relates the sides of a right triangle. The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In our right triangle ABC, where angle B is the right angle, the side adjacent to angle A is AB, and the hypotenuse is AC. We know that AB = 8 and we previously calculated AC = 10 using the Pythagorean theorem. Therefore, to find cos A, we use the formula:

cos(angle) = Adjacent / Hypotenuse

Applying this to angle A in triangle ABC, we have:

cos A = AB / AC

Substitute the known values:

cos A = 8 / 10

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

cos A = 4 / 5

Therefore, the cosine of angle A is 4/5. This value represents the ratio of the adjacent side to the hypotenuse. The cosine function, like the sine function, is a cornerstone of trigonometry. It helps describe the relationships between angles and sides in triangles, and it has widespread applications in various fields. In physics, cosine is often used to calculate components of vectors, particularly in mechanics and electromagnetism. For example, when analyzing the forces acting on an object, the cosine of the angle between the force vector and the horizontal axis gives the horizontal component of the force. In engineering, cosine is crucial in structural analysis, surveying, and many other areas. The cosine function is also fundamental in the study of waves and oscillations, where it describes the behavior of periodic phenomena such as sound waves and alternating current. Understanding how to determine cos A is not only important for trigonometry but also for a wide range of practical applications in science and engineering.

4. Calculating Tan C

The tangent of an angle in a right triangle, denoted as Tan C, is a key trigonometric ratio that connects the lengths of the triangle's legs. Specifically, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In the context of our right triangle ABC, with the right angle at B, we want to find Tan C. The side opposite angle C is AB, and the side adjacent to angle C is BC. We are given that AB = 8 and BC = 6.

The formula for the tangent function is:

tan(angle) = Opposite / Adjacent

Applying this to angle C in triangle ABC, we get:

tan C = AB / BC

Substitute the given values:

tan C = 8 / 6

To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 2:

tan C = 4 / 3

Thus, the tangent of angle C is 4/3. This ratio tells us how the opposite side relates to the adjacent side, providing valuable information about the angle's steepness or slope. The tangent function is widely used in various fields, including surveying, navigation, and engineering. In surveying, it helps in determining angles and distances in land measurement. In navigation, it assists in calculating bearings and courses. In engineering, tangent is crucial in structural analysis and design, especially when dealing with slopes and inclines. For example, engineers use the tangent function to calculate the angle of a ramp or the stability of a structure on a slope. Moreover, the tangent function is essential in calculus and other advanced mathematical studies, where it appears in derivatives and integrals. Therefore, understanding how to calculate Tan C is not only a fundamental trigonometric skill but also a practical tool in many real-world applications.

Conclusion

In conclusion, we have successfully calculated the length of AC and the trigonometric ratios Sin A, Cos A, and Tan C for the given right triangle ABC without using a calculator. By applying the Pythagorean theorem, we found AC = 10. Then, using the definitions of sine, cosine, and tangent, we determined that Sin A = 3/5, Cos A = 4/5, and Tan C = 4/3. These calculations underscore the importance of understanding basic trigonometric principles and their application in solving geometric problems. Mastering these concepts provides a strong foundation for more advanced studies in mathematics, physics, engineering, and other related fields. The ability to perform these calculations manually not only enhances problem-solving skills but also deepens the understanding of the relationships between the sides and angles of right triangles. These skills are essential for various practical applications, from simple navigation to complex structural engineering designs.