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# help

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

Aug 16, 2023

#4
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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   Aug 16, 2023

#3
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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
+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   CPhill Aug 16, 2023
#5
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thanks!

newsss  Aug 16, 2023