Math  /  Trigonometry

QuestionIn TUV\triangle \mathrm{TUV}, the measure of V=90,UT=5,VU=4\angle \mathrm{V}=90^{\circ}, \mathrm{UT}=5, \mathrm{VU}=4, and TV=3\mathrm{TV}=3. What is the value of the cosine of U\angle \mathrm{U} to the nearest hundredth?

Studdy Solution

STEP 1

1. TUV\triangle \mathrm{TUV} is a right triangle with V=90\angle \mathrm{V} = 90^\circ.
2. The sides of the triangle are UT=5\mathrm{UT} = 5, VU=4\mathrm{VU} = 4, and TV=3\mathrm{TV} = 3.

STEP 2

1. Identify the sides relative to U\angle \mathrm{U}.
2. Use the definition of cosine for a right triangle.
3. Calculate the cosine of U\angle \mathrm{U}.
4. Round the result to the nearest hundredth.

STEP 3

Identify the sides relative to U\angle \mathrm{U}:
- The adjacent side to U\angle \mathrm{U} is VU=4\mathrm{VU} = 4. - The hypotenuse is UT=5\mathrm{UT} = 5.

STEP 4

Use the definition of cosine for a right triangle:
cos(U)=Adjacent sideHypotenuse \cos(\angle \mathrm{U}) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}

STEP 5

Substitute the known values into the cosine formula:
cos(U)=45 \cos(\angle \mathrm{U}) = \frac{4}{5}
Calculate the cosine:
cos(U)=0.8 \cos(\angle \mathrm{U}) = 0.8

STEP 6

Round the result to the nearest hundredth:
The value of cos(U)\cos(\angle \mathrm{U}) is already 0.80.8, which is 0.800.80 when rounded to the nearest hundredth.
The value of the cosine of U\angle \mathrm{U} is:
0.80 \boxed{0.80}

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