Exercise 7 Page 655

Draw and label the side lengths of a 30^(∘)-60^(∘)-90^(∘) triangle.

sqrt(3)/2 or approximately 0.87

Practice makes perfect

We want to use a special triangle to express sin 60^(∘) as a fraction and as a decimal to the nearest hundredth. Let's begin with drawing a 30^(∘)-60^(∘)-90^(∘) triangle. If we call the shorter leg of this right triangle x, then the longer leg is xsqrt(3) and the hypotenuse is 2x.

To find the sine of 60^(∘), let's recall one of the trigonometric ratios.

Trigonometric Ratio
If △ ABC is a right triangle with acute ∠ A, then the sine of ∠ A (written sin A) is the ratio of the length of the leg opposite ∠ A to the length of hypotenuse.

Using this definition we can write the formula for sin A. sin A = opposite/hypotenuse We can use the side lengths of our triangle, to write the appropriate ratio for 60^(∘). To do this we will substitute 60^(∘) for A, xsqrt(3) for opposite, and 2x for hypotenuse.

sin A = opposite/hypotenuse

SubstituteValues

Substitute values

sin 60^(∘)=xsqrt(3)/2x

ReduceFrac

a/b=.a /x./.b /x.

sin 60^(∘)=sqrt(3)/2

We found sin 60^(∘) written as a fraction. Now, let's use a calculator to write it as a decimal. Then, we will round it to the nearest hundredth.

sin 60^(∘)=sqrt(3)/2

UseCalc

Use a calculator

sin 60^(∘)=0.86602...