Answered

Find detailed and accurate answers to your questions on IDNLearn.com. Ask anything and receive thorough, reliable answers from our community of experienced professionals.

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

A right triangle [tex]\( ABC \)[/tex] has complementary angles [tex]\( A \)[/tex] and [tex]\( C \)[/tex].

If [tex]\(\sin(A)=\frac{24}{25}\)[/tex], the value of [tex]\(\cos(C)=\ \boxed{}\)[/tex]

If [tex]\(\cos(C)=\frac{20}{29}\)[/tex], the value of [tex]\(\sin(A)=\ \boxed{}\)[/tex]

Sagot :

GinnyAnsweridn
Let's go through the steps to solve the problem:

### Part 1: Finding [tex]\(\cos(C)\)[/tex] given [tex]\(\sin(A)=\frac{24}{25}\)[/tex]

Given: [tex]\(\sin(A) = \frac{24}{25}\)[/tex]

In a right triangle, the sine of angle [tex]\(A\)[/tex] can be described as the ratio of the length of the opposite side to the hypotenuse. This means:
- Opposite side (to angle [tex]\(A\)[/tex]) = 24
- Hypotenuse = 25

To find [tex]\(\cos(C)\)[/tex], we first need to determine the adjacent side. Using the Pythagorean theorem:

[tex]\[ (\text{Adj})^2 + (\text{Opposite})^2 = (\text{Hypotenuse})^2 \][/tex]
[tex]\[ (\text{Adj})^2 + 24^2 = 25^2 \][/tex]
[tex]\[ (\text{Adj})^2 + 576 = 625 \][/tex]
[tex]\[ (\text{Adj})^2 = 625 - 576 \][/tex]
[tex]\[ (\text{Adj})^2 = 49 \][/tex]
[tex]\[ \text{Adj} = \sqrt{49} = 7 \][/tex]

So, the adjacent side (to angle [tex]\(A\)[/tex]) is 7.

Since angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary in a right triangle, [tex]\(\sin(A) = \cos(C)\)[/tex].

Therefore, [tex]\(\cos(C) = \frac{\text{Adj}}{\text{Hypotenuse}} = \frac{7}{25} = 0.28\)[/tex]

[tex]\(\cos(C) = \boxed{0.28}\)[/tex]


### Part 2: Finding [tex]\(\sin(A)\)[/tex] given [tex]\(\cos(C)=\frac{20}{29}\)[/tex]

Given: [tex]\(\cos(C) = \frac{20}{29}\)[/tex]

In a right triangle, the cosine of angle [tex]\(C\)[/tex] can be described as the ratio of the length of the adjacent side to the hypotenuse. This means:
- Adjacent side (to angle [tex]\(C\)[/tex]) = 20
- Hypotenuse = 29

To find [tex]\(\sin(A)\)[/tex], we first need to determine the opposite side. Using the Pythagorean theorem:

[tex]\[ (\text{Opposite})^2 + (\text{Adj})^2 = (\text{Hypotenuse})^2 \][/tex]
[tex]\[ (\text{Opposite})^2 + 20^2 = 29^2 \][/tex]
[tex]\[ (\text{Opposite})^2 + 400 = 841 \][/tex]
[tex]\[ (\text{Opposite})^2 = 841 - 400 \][/tex]
[tex]\[ (\text{Opposite})^2 = 441 \][/tex]
[tex]\[ \text{Opposite} = \sqrt{441} = 21 \][/tex]

So, the opposite side (to angle [tex]\(C\)[/tex]) is 21.

Since angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are complementary in a right triangle, [tex]\(\cos(C) = \sin(A)\)[/tex].

Therefore, [tex]\(\sin(A) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{21}{29} \approx 0.7241379310344828\)[/tex]

[tex]\(\sin(A) = \boxed{0.7241379310344828}\)[/tex]

So, we have:
- [tex]\(\cos(C) = 0.28\)[/tex]
- [tex]\(\sin(A) = 0.7241379310344828\)[/tex]
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.