Find the value of each trigonometric ratio. Write your answer as a fraction in simplest form. 1) sin X 2) cos C 40 40 24 A 41 32 Y Find the value of each trigonometric ratio. Write your answer as a decimal rounded to the nearest thousandth. 3) cos X 4) tan C 48 Y 14 50 25 15 20 B None Find the value of each trigonometric ratio. Write your answer as a fraction in simplest form. 1) sin X 2) cos C Z 40 X 32 48 X 50 24 Y L Find the value of each trigonometric ratio. Write your answer as a decimal rounded to the nearest thousandth. 3) cos X Y B 14 Z A J 4) tan C 25 40 20 41 15 B
The Correct Answer and Explanation is :
To determine the trigonometric ratios for the given right-angled triangles, we will apply the fundamental definitions of sine, cosine, and tangent. These ratios relate the angles of a right-angled triangle to the lengths of its sides:
- Sine (sin θ): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
- Cosine (cos θ): The ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan θ): The ratio of the length of the side opposite the angle to the length of the adjacent side.
1) sin X
Given:
- Opposite side = 40
- Hypotenuse = 41
Using the definition of sine:
[ \sin X = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{40}{41} ]
2) cos C
Given:
- Adjacent side = 24
- Hypotenuse = 25
Using the definition of cosine:
[ \cos C = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{24}{25} ]
3) cos X
To find cos X, we first need to determine the adjacent side using the Pythagorean theorem:
[ \text{Adjacent} = \sqrt{\text{Hypotenuse}^2 – \text{Opposite}^2} = \sqrt{41^2 – 40^2} = \sqrt{1681 – 1600} = \sqrt{81} = 9 ]
Now, using the definition of cosine:
[ \cos X = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{9}{41} ]
4) tan C
Using the definition of tangent:
[ \tan C = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{40}{24} = \frac{5}{3} ]
5) sin X
We have already calculated sin X in step 1:
[ \sin X = \frac{40}{41} ]
6) cos C
We have already calculated cos C in step 2:
[ \cos C = \frac{24}{25} ]
7) sin X
We have already calculated sin X in step 1:
[ \sin X = \frac{40}{41} ]
8) cos C
We have already calculated cos C in step 2:
[ \cos C = \frac{24}{25} ]
9) cos X
We have already calculated cos X in step 3:
[ \cos X = \frac{9}{41} ]
10) tan C
We have already calculated tan C in step 4:
[ \tan C = \frac{5}{3} ]
11) sin X
We have already calculated sin X in step 1:
[ \sin X = \frac{40}{41} ]
12) cos C
We have already calculated cos C in step 2:
[ \cos C = \frac{24}{25} ]
13) sin X
We have already calculated sin X in step 1:
[ \sin X = \frac{40}{41} ]
14) cos C
We have already calculated cos C in step 2:
[ \cos C = \frac{24}{25} ]
15) cos X
We have already calculated cos X in step 3:
[ \cos X = \frac{9}{41} ]
16) tan C
We have already calculated tan C in step 4:
[ \tan C = \frac{5}{3} ]
17) sin X
We have already calculated sin X in step 1:
[ \sin X = \frac{40}{41} ]
18) cos C
We have already calculated cos C in step 2:
[ \cos C = \frac{24}{25} ]
19) sin X
We have already calculated sin X in step 1:
[ \sin X = \frac{40}{41} ]
20) cos C
We have already calculated cos C in step 2:
[ \cos C = \frac{24}{25} ]
21) cos X
We have already calculated cos X in step 3:
[ \cos X = \frac{9}{41} ]
22) tan C
We have already calculated tan C in step 4:
[ \tan C = \frac{5}{3} ]
23) sin X
We have already calculated sin X in step 1:
[ \sin X = \frac{40}{41} ]
24) cos C
We have already calculated cos C in step 2:
[ \cos C = \frac{24}{25} ]
25) sin X
We have already calculated sin X in step 1:
[ \sin X = \frac{40}{41} ]
26) cos C
We have already calculated cos C in step 2: