Cumulative Assessment - 9. Right Triangles and Trigonometry

We will determine the values of all ten expressions. First let's take care of sine, cosine, and tangent ratios. Below we recall how to calculate these trigonometric ratios.

Let's take a look at the given right triangle.

We will calculate sine, cosine, and tangent ratios for angles X and Z. Let's start with X. &sinX=3sqrt(3)/6=sqrt(3)/2 [0.8em] &cosX=3/6=1/2 [0.8em] &tanX=3sqrt(3)/3=sqrt(3) Now we will calculate these ratios for Z. &sinZ=3/6=1/2 [0.8em] &cosZ=3sqrt(3)/6=sqrt(3)/2 [0.8em] &tanZ=3/3sqrt(3)=sqrt(3)/3 Next we will calculate the four remaining ratios. &XY/XZ=3/6=1/2 [0.8em] &YZ/XZ=3sqrt(3)/6=sqrt(3)/2 [0.8em] &XY/YZ=3/3sqrt(3)=sqrt(3)/3 [0.8em] &YZ/XY=3sqrt(3)/3=sqrt(3) Finally we can create as many true equations as possible. & sinX=cosZ & sinX=YZ/XZ [0.8em] & cosX=sinZ & cosX=XY/XZ [0.8em] & tanX=YZ/XY & sinZ=XY/XZ [0.8em] & cosZ=YZ/XZ & tanZ=XY/YZ Please note that since the equality is symmetric, if we consider an equality a=b we do not consider an equality b=a.

Extra

Extra Explanation

We do not have to calculate the values of all ten expressions. Instead, we can use the properties of sine, cosine, and tangent ratios. Let's take a look at the following two equations. & sinX=cosZ & cosX=sinZ They are true because Z is a complement of X, so Z=90^(∘)-X. This allows us to use the formulas for sine and cosine of complementary angles. Now we will consider the remaining equations. & sinX=YZ/XZ & cosX=XY/XZ [0.8em] & tanX=YZ/XY & sinZ=XY/XZ [0.8em] & cosZ=YZ/XZ & tanZ=XY/YZ All of these equations result from the definitions of sine, cosine, and tangent ratios.

Exercise 5 Page 525

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