Exercise 5 Page 892

Consider the United States 34-star flag. We are told that for this flag tan θ = 31.551 and we want to find sin θ.

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The diagonal splits the flag in two right triangles, and θ is an acute angle in one of them. In order to find the value of sin θ, we will the definitions of the trigonometric ratios for tangent and sine. tanθ=opposite/adjacent sinθ=opposite/hypotenuse From the given information, we know that the ratio of the length of the opposite leg to the length of the adjacent leg is 31.5 51. Please note that these are not the lengths of the legs but rather the ratio between them. To make this ratio a bit easier to work with, let's rewrite it without the decimal value. 31.5/51* 2/2=63/102 Next, we will use the Pythagorean Theorem to find the hypotenuse. Once again, this will not be the actual length of the hypotenuse but rather the length required to fit the ratio that we already have. This works because of the concept of similar triangles.

Finally, we can substitute the values that we have for opposite and hypotenuse into the definition of sine to find the ratio of the length of the opposite side to the length of the hypotenuse. Because we will have a square root in the denominator, we need to rationalize the denominator.