Question

Hello I need help with question 11 ! I will give you a great rating! Please help, I’m not sure how to do this. Also this is not a quiz this is practice

Answer 1

Given the ΔABC as shown below:

To evaluate the value of x, y, and z,

Let's begin with z.

Step 1: In the ΔBCD, identify the sides.

Thus, in the triangle BCD,

`\begin{gathered} BC\Rightarrow hypotenuse \\ BD\Rightarrow opposite \\ DC\Rightarrow adjacent \end{gathered}`

Step 2: Evaluate the value of z, using trigonometric ratios.

From trigonometric ratios,

`\begin{gathered} sin\text{ }\theta=(opposite)/(hypotenuse) \\ cos\text{ }\theta=(adjacent)/(hypotenuse) \\ tan\text{ }\theta=(opposite)/(adjacent) \end{gathered}`

Thus, we have

`\begin{gathered} cos\text{ }\theta=(adjacent)/(hypotenuse) \\ where \\ \theta\Rightarrow\angle C=30 \\ adjacent\Rightarrow DC=z \\ hypotenuse\Rightarrow BC=24 \\ thus, \\ cos\text{ 30=}(z)/(24) \\ cross-multiply, \\ z=24* cos\text{ 30} \\ =24*(√(3))/(2) \\ \Rightarrow z=12√(3) \end{gathered}`

To evaluate the values of x and y, we need to first evaluate the value of BD.

Thus,

`\begin{gathered} \sin\theta=(opposite)/(hypotenuse) \\ where \\ \theta=30 \\ opposite=BD \\ hypotenuse=BC=24 \\ thus, \\ \sin30=(BD)/(24) \\ cross-multiply, \\ BD=24* sin\text{ 30} \\ =24*(1)/(2) \\ \Rightarrow BD=12 \end{gathered}`

Thus, to evaluate the value of x,

step 1: In the ΔABD, identify the sides of the triangle.

Thus, in the triangle ABD,

`\begin{gathered} hypotenuse\Rightarrow AB \\ opposite\Rightarrow AD \\ adjacent\Rightarrow BD \end{gathered}`

Step 2: Evaluate the value of x, using trigonometric ratios.

From trigonometric ratios,

`\begin{gathered} cos\text{ }\theta=(adjacent)/(hyptenuse) \\ where \\ \theta\Rightarrow\angle B=45 \\ adjacent\Rightarrow BD=12 \\ hypotenuse\Rightarrow AB=x \\ thus, \\ cos\text{ 45 = }(12)/(x) \\ cross-multiply, \\ x* cos\text{ 45 = 12} \\ \Rightarrow x*(√(2))/(2)=12 \\ divide\text{ both sides by }(√(2))/(2), \\ x=12*(2)/(√(2))=(24)/(√(2)) \\ rationalize\text{ the denominator of the surd,} \\ x=(24√(2))/(√(2))*(√(2))/(√(2))=(24√(2))/(2) \\ \Rightarrow x=12√(2) \end{gathered}`

To evaluate the value of y,

In the triangle ABD, using trigonometric ratios,

`\begin{gathered} \tan\theta=(opposite)/(adjacent) \\ where \\ \theta=45 \\ opposite\Rightarrow AD=y \\ adjacent\Rightarrow BD=12 \\ thus, \\ \tan45=(y)/(12) \\ cross-multiply, \\ y=12*\tan45 \\ =12*1 \\ \Rightarrow y=12 \end{gathered}`

Hence, the values of x, y and z are

`\begin{gathered} x=12√(2) \\ y=12 \\ z=12√(3) \end{gathered}`