Select The Correct Answer.In Right Triangle \[$ABC\$\], \[$\angle A\$\] And \[$\angle B\$\] Are Complementary Angles, And \[$\sin A = \frac{8}{9}\$\]. What Is \[$\cos B\$\]?A. \[$\frac{8

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Select the Correct Answer: Trigonometry in Right Triangle ABC

In this article, we will explore the concept of trigonometry in a right triangle, specifically focusing on the relationship between complementary angles and their trigonometric ratios. We will use the given information about angle A and its sine value to find the cosine value of angle B.

Understanding Complementary Angles

Complementary angles are two angles whose sum is 90 degrees. In the given right triangle ABC, angles A and B are complementary angles, meaning that ∠A + ∠B = 90°.

Recalling Trigonometric Ratios

Trigonometric ratios are used to describe the relationships between the angles and side lengths of a right triangle. The sine, cosine, and tangent 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.

Given Information

We are given that sin A = 8/9. This means that the ratio of the length of the side opposite angle A to the length of the hypotenuse is 8/9.

Finding Cos B

To find cos B, we need to use the relationship between the sine and cosine ratios. Since angles A and B are complementary, we can use the following identity:

cos B = sin A

This is because the sine of an angle is equal to the cosine of its complementary angle.

Substituting the Given Value

We are given that sin A = 8/9. Substituting this value into the equation above, we get:

cos B = 8/9

Conclusion

In this article, we used the given information about angle A and its sine value to find the cosine value of angle B. We recalled the trigonometric ratios and used the relationship between the sine and cosine ratios to find cos B. The final answer is cos B = 8/9.

Additional Information

  • Complementary Angles: Two angles whose sum is 90 degrees.
  • Trigonometric Ratios: The sine, cosine, and tangent ratios are used to describe the relationships between the angles and side lengths of a right triangle.
  • 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.

Frequently Asked Questions

  • What are complementary angles?
    • Complementary angles are two angles whose sum is 90 degrees.
  • What is the relationship between the sine and cosine ratios?
    • The sine of an angle is equal to the cosine of its complementary angle.
  • How do you find cos B if you know sin A?
    • Use the relationship cos B = sin A to find cos B.

References

  • Trigonometry: A branch of mathematics that deals with the relationships between the angles and side lengths of triangles.
  • Right Triangle: A triangle with one right angle (90 degrees).
  • Hypotenuse: The side opposite the right angle in a right triangle.
  • Adjacent Side: The side next to the angle in a right triangle.
  • Opposite Side: The side opposite the angle in a right triangle.
    Select the Correct Answer: Trigonometry in Right Triangle ABC - Q&A

In our previous article, we explored the concept of trigonometry in a right triangle, specifically focusing on the relationship between complementary angles and their trigonometric ratios. We used the given information about angle A and its sine value to find the cosine value of angle B. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q: What are complementary angles?

A: Complementary angles are two angles whose sum is 90 degrees. In the given right triangle ABC, angles A and B are complementary angles, meaning that ∠A + ∠B = 90°.

Q: What is the relationship between the sine and cosine ratios?

A: The sine of an angle is equal to the cosine of its complementary angle. This means that sin A = cos B.

Q: How do you find cos B if you know sin A?

A: Use the relationship cos B = sin A to find cos B. This is because the sine and cosine ratios are equal for complementary angles.

Q: What is the formula for finding the cosine of an angle?

A: The formula for finding the cosine of an angle is cos A = adjacent side / hypotenuse.

Q: What is the formula for finding the sine of an angle?

A: The formula for finding the sine of an angle is sin A = opposite side / hypotenuse.

Q: What is the tangent of an angle?

A: 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. It is calculated as tan A = opposite side / adjacent side.

Q: How do you use the Pythagorean theorem in trigonometry?

A: The Pythagorean theorem states that a² + b² = c², where a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse. This theorem can be used to find the length of the hypotenuse or the length of one of the legs.

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

A: The sine ratio is the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine ratio is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: Can you provide examples of how to use the sine and cosine ratios?

A: Yes, here are a few examples:

  • Find the sine of angle A if the length of the side opposite angle A is 3 and the length of the hypotenuse is 5.
    • sin A = opposite side / hypotenuse = 3 / 5 = 0.6
  • Find the cosine of angle B if the length of the side adjacent to angle B is 4 and the length of the hypotenuse is 5.
    • cos B = adjacent side / hypotenuse = 4 / 5 = 0.8

Conclusion

In this Q&A article, we provided answers to common questions about trigonometry in right triangles. We covered topics such as complementary angles, the relationship between the sine and cosine ratios, and how to use the Pythagorean theorem in trigonometry. We also provided examples of how to use the sine and cosine ratios to find the values of angles in a right triangle.

Additional Information

  • Trigonometry: A branch of mathematics that deals with the relationships between the angles and side lengths of triangles.
  • Right Triangle: A triangle with one right angle (90 degrees).
  • Hypotenuse: The side opposite the right angle in a right triangle.
  • Adjacent Side: The side next to the angle in a right triangle.
  • Opposite Side: The side opposite the angle in a right triangle.

References

  • Trigonometry: A branch of mathematics that deals with the relationships between the angles and side lengths of triangles.
  • Right Triangle: A triangle with one right angle (90 degrees).
  • Hypotenuse: The side opposite the right angle in a right triangle.
  • Adjacent Side: The side next to the angle in a right triangle.
  • Opposite Side: The side opposite the angle in a right triangle.