Recall of Sine and Cosine Definitions

Oct 13, 2025 by talynt.org 32 views

In A ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine: (i) sin A, cos A (ii) sin C, Cos C

In this problem, we are given a right-angled triangle ABC with AB = 24 cm and BC = 7 cm. We need to determine the values of sin A, cos A, sin C, and cos C. To do this, we will use the definitions of sine and cosine in a right-angled triangle.

Recall of Sine and Cosine Definitions

In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. 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.

Step 1: Determine the Length of the Hypotenuse

To determine the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Pythagorean Theorem

a^2 + b^2 = c^2

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

Applying the Pythagorean Theorem

In this case, we have a = 24 cm and b = 7 cm. Plugging these values into the Pythagorean theorem, we get:

24^2 + 7^2 = c^2 576 + 49 = c^2 625 = c^2

Taking the square root of both sides, we get:

c = √625 c = 25 cm

Step 2: Determine sin A and cos A

Now that we have the length of the hypotenuse, we can determine the values of sin A and cos A.

Sine of A

The sine of A is defined as the ratio of the length of the side opposite A to the length of the hypotenuse. In this case, the side opposite A is BC, which has a length of 7 cm. The length of the hypotenuse is 25 cm.

sin A = opposite / hypotenuse = 7 / 25 = 0.28

Cosine of A

The cosine of A is defined as the ratio of the length of the side adjacent to A to the length of the hypotenuse. In this case, the side adjacent to A is AB, which has a length of 24 cm. The length of the hypotenuse is 25 cm.

cos A = adjacent / hypotenuse = 24 / 25 = 0.96

Step 3: Determine sin C and cos C

Now that we have the values of sin A and cos A, we can determine the values of sin C and cos C.

Sine of C

The sine of C is defined as the ratio of the length of the side opposite C to the length of the hypotenuse. In this case, the side opposite C is AB, which has a length of 24 cm. The length of the hypotenuse is 25 cm.

sin C = opposite / hypotenuse = 24 / 25 = 0.96

Cosine of C

The cosine of C is defined as the ratio of the length of the side adjacent to C to the length of the hypotenuse. In this case, the side adjacent to C is BC, which has a length of 7 cm. The length of the hypotenuse is 25 cm.

cos C = adjacent / hypotenuse = 7 / 25 = 0.28

In this problem, we have determined the values of sin A, cos A, sin C, and cos C in a right-angled triangle ABC with AB = 24 cm and BC = 7 cm. We have used the definitions of sine and cosine in a right-angled triangle and the Pythagorean theorem to determine the values of sin A, cos A, sin C, and cos C.

Table of Results

Angle	sin	cos
A	0.28	0.96
C	0.96	0.28

Note

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right-angled 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.

Q: How do I use the Pythagorean theorem to find the length of the hypotenuse?

A: To use the Pythagorean theorem to find the length of the hypotenuse, you need to know the lengths of the other two sides. You can then plug these values into the formula: c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Q: What is the difference between sine and cosine?

A: Sine and cosine are two trigonometric functions that describe the relationships between the angles and sides of a right-angled triangle. Sine is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, while cosine is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How do I use the definitions of sine and cosine to find the values of sin A and cos A?

A: To use the definitions of sine and cosine to find the values of sin A and cos A, you need to know the lengths of the sides of the triangle. You can then plug these values into the formulas: sin A = opposite / hypotenuse and cos A = adjacent / hypotenuse.

Q: What is the relationship between sin A and cos A?

A: The relationship between sin A and cos A is that they are complementary angles. This means that sin A + cos A = 1.

Q: How do I use the values of sin A and cos A to find the values of sin C and cos C?

A: To use the values of sin A and cos A to find the values of sin C and cos C, you need to know the relationship between the angles A and C. Since A and C are complementary angles, you can use the formulas: sin C = cos A and cos C = sin A.

Q: What is the relationship between sin C and cos C?

A: The relationship between sin C and cos C is that they are complementary angles. This means that sin C + cos C = 1.

Q: Can I use the values of sin A and cos A to find the values of sin B and cos B?

A: Yes, you can use the values of sin A and cos A to find the values of sin B and cos B. Since A and B are complementary angles, you can use the formulas: sin B = cos A and cos B = sin A.

Q: What is the relationship between sin B and cos B?

A: The relationship between sin B and cos B is that they are complementary angles. This means that sin B + cos B = 1.

In this article, we have answered some of the most frequently asked questions about trigonometry and the Pythagorean theorem. We have covered topics such as the definitions of sine and cosine, the Pythagorean theorem, and the relationships between the angles and sides of a right-angled triangle. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of these important concepts.