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.1. If Sin ⁡ ( A ) = 24 25 \sin (A)=\frac{24}{25} Sin ( A ) = 25 24 ​ , The Value Of Cos ⁡ ( C ) = \cos (C)= Cos ( C ) =

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Introduction

In the world of mathematics, particularly in trigonometry, understanding the relationships between angles and their corresponding trigonometric ratios is crucial. One such concept is complementary angles, which are pairs of angles that add up to 90 degrees. In this article, we will delve into the world of right triangles and explore how complementary angles A and C relate to the sine and cosine functions.

What are Complementary Angles?

Complementary angles are pairs of angles that add up to 90 degrees. In the context of a right triangle, the two acute angles (angles less than 90 degrees) are complementary. This means that if we have a right triangle ABC with angle A and angle C, then A + C = 90 degrees.

Understanding Sine and Cosine

Sine and cosine are two fundamental trigonometric ratios that are used to describe the relationships between the angles and the side lengths of a right triangle. The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse (the side opposite the right angle). The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Given Information

We are given that sin⁡(A)=2425\sin (A)=\frac{24}{25} in a right triangle ABC with complementary angles A and C. We need to find the value of cos⁡(C)\cos (C).

Using Complementary Angles to Find Cosine of C

Since A and C are complementary angles, we know that A + C = 90 degrees. This means that C = 90 - A. We can use this relationship to find the value of cos⁡(C)\cos (C).

Using the definition of cosine, we know that cos⁡(C)=adjacent sidehypotenuse\cos (C) = \frac{\text{adjacent side}}{\text{hypotenuse}}. Since C is an acute angle, the adjacent side is the side opposite angle A. Let's call this side "a". The hypotenuse is the side opposite the right angle, which we'll call "h".

We can use the Pythagorean theorem to relate the side lengths of the triangle:

a2+b2=h2a^2 + b^2 = h^2

where b is the side opposite angle C.

Since sin⁡(A)=2425\sin (A)=\frac{24}{25}, we know that ah=2425\frac{a}{h}=\frac{24}{25}. We can use this to find the value of cos⁡(C)\cos (C).

Finding the Value of Cosine of C

Using the definition of cosine, we have:

cos⁡(C)=adjacent sidehypotenuse=ah\cos (C) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{a}{h}

We know that ah=2425\frac{a}{h}=\frac{24}{25}, so we can substitute this into the equation:

cos⁡(C)=2425\cos (C) = \frac{24}{25}

However, we need to find the value of cos⁡(C)\cos (C), not cos⁡(A)\cos (A). Since C = 90 - A, we can use the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A) to find the value of cos⁡(C)\cos (C).

Using the Identity to Find Cosine of C

Using the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A), we have:

cos⁡(C)=cos⁡(90−A)=sin⁡(A)=2425\cos (C) = \cos (90 - A) = \sin (A) = \frac{24}{25}

Therefore, the value of cos⁡(C)\cos (C) is 2425\boxed{\frac{24}{25}}.

Conclusion

In this article, we explored the concept of complementary angles in right triangles and how they relate to the sine and cosine functions. We used the given information to find the value of cos⁡(C)\cos (C), which is 2425\boxed{\frac{24}{25}}. This demonstrates the importance of understanding the relationships between angles and their corresponding trigonometric ratios in mathematics.

Key Takeaways

  • Complementary angles are pairs of angles that add up to 90 degrees.
  • The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
  • The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
  • Using the Pythagorean theorem and trigonometric identities, we can find the value of cos⁡(C)\cos (C) given the value of sin⁡(A)\sin (A).

Practice Problems

  1. If sin⁡(A)=1617\sin (A)=\frac{16}{17}, find the value of cos⁡(C)\cos (C).
  2. If cos⁡(A)=910\cos (A)=\frac{9}{10}, find the value of sin⁡(C)\sin (C).
  3. If sin⁡(A)=2021\sin (A)=\frac{20}{21}, find the value of cos⁡(C)\cos (C).

Solutions

  1. If sin⁡(A)=1617\sin (A)=\frac{16}{17}, we can use the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A) to find the value of cos⁡(C)\cos (C):

cos⁡(C)=cos⁡(90−A)=sin⁡(A)=1617\cos (C) = \cos (90 - A) = \sin (A) = \frac{16}{17}

  1. If cos⁡(A)=910\cos (A)=\frac{9}{10}, we can use the identity sin⁡(A)=1−cos⁡2(A)\sin (A) = \sqrt{1 - \cos^2 (A)} to find the value of sin⁡(A)\sin (A):

sin⁡(A)=1−cos⁡2(A)=1−(910)2=310\sin (A) = \sqrt{1 - \cos^2 (A)} = \sqrt{1 - \left(\frac{9}{10}\right)^2} = \frac{3}{10}

We can then use the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A) to find the value of cos⁡(C)\cos (C):

cos⁡(C)=cos⁡(90−A)=sin⁡(A)=310\cos (C) = \cos (90 - A) = \sin (A) = \frac{3}{10}

  1. If sin⁡(A)=2021\sin (A)=\frac{20}{21}, we can use the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A) to find the value of cos⁡(C)\cos (C):

Q: What are complementary angles?

A: Complementary angles are pairs of angles that add up to 90 degrees. In the context of a right triangle, the two acute angles (angles less than 90 degrees) are complementary.

Q: How do complementary angles relate to the sine and cosine functions?

A: Complementary angles are related to the sine and cosine functions through the Pythagorean theorem and trigonometric identities. Specifically, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Q: How can I use the Pythagorean theorem to find the value of cosine of C?

A: To find the value of cosine of C, you can use the Pythagorean theorem to relate the side lengths of the triangle. Specifically, you can use the equation a2+b2=h2a^2 + b^2 = h^2 to find the value of the adjacent side (a) and the hypotenuse (h).

Q: What is the relationship between the sine of A and the cosine of C?

A: The sine of A and the cosine of C are related through the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A). This means that the cosine of C is equal to the sine of A.

Q: How can I use the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A) to find the value of cosine of C?

A: To find the value of cosine of C, you can use the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A) and substitute the value of the sine of A into the equation. This will give you the value of the cosine of C.

Q: What are some common mistakes to avoid when working with complementary angles?

A: Some common mistakes to avoid when working with complementary angles include:

  • Failing to recognize that complementary angles add up to 90 degrees
  • Confusing the sine and cosine functions
  • Not using the Pythagorean theorem to relate the side lengths of the triangle
  • Not using the identity cos⁡(90−A)=sin⁡(A)\cos (90 - A) = \sin (A) to find the value of cosine of C

Q: How can I practice working with complementary angles?

A: To practice working with complementary angles, you can try the following:

  • Work through examples and exercises that involve complementary angles
  • Use online resources or math software to visualize and explore the relationships between complementary angles
  • Practice using the Pythagorean theorem and trigonometric identities to find the value of cosine of C
  • Try solving problems that involve complementary angles in different contexts, such as physics or engineering.

Q: What are some real-world applications of complementary angles?

A: Complementary angles have many real-world applications, including:

  • Physics: Complementary angles are used to describe the relationships between the angles and side lengths of triangles in physics problems.
  • Engineering: Complementary angles are used to design and analyze the structures and systems of buildings, bridges, and other infrastructure.
  • Computer Science: Complementary angles are used in computer graphics and game development to create 3D models and animations.
  • Navigation: Complementary angles are used in navigation systems, such as GPS, to determine the user's location and direction.

Q: How can I use complementary angles to solve problems in other areas of mathematics?

A: Complementary angles can be used to solve problems in other areas of mathematics, including:

  • Algebra: Complementary angles can be used to solve systems of equations and linear algebra problems.
  • Geometry: Complementary angles can be used to solve problems involving the properties and relationships of geometric shapes.
  • Trigonometry: Complementary angles can be used to solve problems involving the relationships between the angles and side lengths of triangles.

Q: What are some common misconceptions about complementary angles?

A: Some common misconceptions about complementary angles include:

  • Thinking that complementary angles are always 90 degrees
  • Confusing the sine and cosine functions
  • Not recognizing that complementary angles are related to the Pythagorean theorem and trigonometric identities
  • Not understanding the relationship between the sine of A and the cosine of C.