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In triangle XYZ, we have angle X = 90 and tan Z = 3. What is cos Z?

 Aug 16, 2023

Best Answer 

 #4
avatar+129243 
+1

tan z  =   opp / adj  =  3 /  1

 

The hypotenuse =  sqrt [ 3^1 + 1^1 ] =  sqrt (10)

 

cos z  = adj / hyp  =    1 /sqrt (10)  =  sqrt (10) / 10

 

cool cool cool

 Aug 16, 2023
 #3
avatar+121 
+1

In a right triangle XYZ where angle X is 90 degrees, we can use the information provided to find the cosine of angle Z.

Given that tan Z = 3, we know that:

\[ \tan Z = \frac{\text{opposite}}{\text{adjacent}} = \frac{YZ}{XZ} = 3.\]

Since triangle XYZ is a right triangle, we can use the Pythagorean theorem:

\[ XY^2 + YZ^2 = XZ^2.\]

Since angle X is 90 degrees, we have:

\[ XZ^2 = XY^2 + YZ^2 = YZ^2 + YZ^2 = 2YZ^2.\]

Solving for YZ:

\[ YZ^2 = \frac{XZ^2}{2}.\]

Since we know the value of YZ/XZ (which is the tangent of angle Z):

\[ \tan Z = \frac{YZ}{XZ} = 3.\]

Squaring both sides:

\[ YZ^2 = 9XZ^2.\]

Now, substituting the value of YZ^2 from the Pythagorean theorem:

\[ 9XZ^2 = \frac{XZ^2}{2}.\]

Solving for XZ:

\[ \frac{XZ^2}{2} = 9XY^2.\]

\[ \cos Z = \boxed{\frac12}.\]

 Aug 16, 2023
 #4
avatar+129243 
+1
Best Answer

tan z  =   opp / adj  =  3 /  1

 

The hypotenuse =  sqrt [ 3^1 + 1^1 ] =  sqrt (10)

 

cos z  = adj / hyp  =    1 /sqrt (10)  =  sqrt (10) / 10

 

cool cool cool

CPhill Aug 16, 2023
 #5
avatar+46 
0

thanks!

newsss  Aug 16, 2023

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