Type The Correct Answer In Each Box. Use Numerals Instead Of Words. If Necessary, Use $/$ For The Fraction Bar(s).A Right Triangle ABC Has Complementary Angles A And C.If Sin ⁡ ( A ) = 24 25 \sin(A) = \frac{24}{25} Sin ( A ) = 25 24 , The Value Of Cos ⁡ ( C ) = \cos(C) = Cos ( C ) =

Mar 12, 2025 by ADMIN 288 views

**Complementary Angles and Trigonometric Functions**

Understanding Complementary Angles

In a right triangle, complementary angles are two angles whose sum is 90 degrees. In the given triangle ABC, angles A and C are complementary angles. This means that angle A + angle C = 90 degrees.

Trigonometric Functions

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

Sine (sin): 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 side opposite the right angle).
Cosine (cos): 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.
Tangent (tan): 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.

Given Information

In the given triangle ABC, we are given that $\sin(A) = \frac{24}{25}$ . We need to find the value of $\cos(C)$ .

Using Complementary Angles

Since angles A and C are complementary angles, we can write:

\sin(A) = \cos(C)

This is because the sine of angle A is equal to the cosine of angle C.

Finding the Value of $\cos(C)$

We are given that $\sin(A) = \frac{24}{25}$ . We can use this information to find the value of $\cos(C)$ .

Since $\sin(A) = \cos(C)$ , we can write:

\cos(C) = \sin(A) = \frac{24}{25}

Therefore, the value of $\cos(C)$ is $\frac{24}{25}$ .

Conclusion

In this article, we used the concept of complementary angles and trigonometric functions to find the value of $\cos(C)$ in a right triangle ABC. We were given that $\sin(A) = \frac{24}{25}$ and we used this information to find the value of $\cos(C)$ . We found that $\cos(C) = \frac{24}{25}$ .

Key Takeaways

Complementary angles are two angles whose sum is 90 degrees.
Trigonometric functions are used to describe the relationships between the angles and side lengths of a right triangle.
The sine, cosine, and tangent functions are the most commonly used trigonometric functions.
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.
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.

Frequently Asked Questions

What are complementary angles?
- Complementary angles are two angles whose sum is 90 degrees.
What are trigonometric functions?
- Trigonometric functions are used to describe the relationships between the angles and side lengths of a right triangle.
What is the sine of an angle?
- 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.
What is the cosine of an angle?
- 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.
What is the tangent of an angle?
- 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.
  Complementary Angles and Trigonometric Functions: Q&A =====================================================

Q: What are complementary angles?

A: Complementary angles are two angles whose sum is 90 degrees. In the context of a right triangle, complementary angles are two angles that add up to 90 degrees.

Q: What is the relationship between sine and cosine?

A: The sine and cosine functions are related in that they are equal for complementary angles. This means that if $\sin(A) = \frac{24}{25}$ , then $\cos(C) = \frac{24}{25}$ , where A and C are complementary angles.

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) = \frac{\text{opposite}}{\text{hypotenuse}}

Where "opposite" is the length of the side opposite the angle, and "hypotenuse" is the length of the side opposite the right angle.

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) = \frac{\text{adjacent}}{\text{hypotenuse}}

Where "adjacent" is the length of the side adjacent to the angle, and "hypotenuse" is the length of the side opposite the right angle.

Q: What is the tangent of an angle?

A: 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. The formula for finding the tangent of an angle is:

\tan(A) = \frac{\text{opposite}}{\text{adjacent}}

Q: How do you find the value of $\cos(C)$ if you know the value of $\sin(A)$ ?

A: If you know the value of $\sin(A)$ , you can find the value of $\cos(C)$ by using the relationship between sine and cosine for complementary angles. Since $\sin(A) = \cos(C)$ , you can write:

\cos(C) = \sin(A)

Therefore, if you know the value of $\sin(A)$ , you can find the value of $\cos(C)$ by substituting the value of $\sin(A)$ into the equation.

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

A: The sine and cosine functions are both used to describe the relationships between the angles and side lengths of a right triangle. However, the sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: Can you give an example of how to use the sine and cosine functions to solve a problem?

A: Yes, here is an example:

Suppose you have a right triangle with an angle A and a side opposite the angle that is 24 units long. The hypotenuse of the triangle is 25 units long. You want to find the value of $\sin(A)$ .

Using the formula for the sine function, you can write:

\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{24}{25}

Therefore, the value of $\sin(A)$ is $\frac{24}{25}$ .

Q: What is the significance of the tangent function?

A: The tangent function is used to describe the relationships between the angles and side lengths of a right triangle. It is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The tangent function is useful for solving problems involving right triangles, such as finding the length of a side or the measure of an angle.

Q: Can you give an example of how to use the tangent function to solve a problem?