To find a decimal approximation for $\sin 51^\circ 42'$, we need to first convert the angle from degrees and minutes to a decimal form in degrees.

Step 1: Convert the Angle

1. The angle is given as $51^\circ 42'$.
2. Convert $42'$ (minutes) into degrees by dividing by 60: $42' = \frac{42}{60} = 0.7^\circ$
3. Add this to $51^\circ$: $51^\circ 42' = 51 + 0.7 = 51.7^\circ$

Step 2: Calculate the Sine

Using a calculator, find the sine of $51.7^\circ$: $\sin 51.7^\circ \approx 0.78625322$

Answer

$\sin 51^\circ 42' \approx 0.78625322$

Let me know if you want more details on the steps or have questions!

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1. What is the decimal approximation of $\cos 51^\circ 42'$?
2. How do you convert minutes to a decimal in trigonometric calculations?
3. What is the sine of an angle in radians if given $51^\circ 42'$?
4. Why is it useful to round trigonometric values to eight decimal places?
5. How would you find $\sin(51^\circ + 42')$ without converting to a decimal?

Tip: Converting minutes to decimal form makes trigonometric calculations easier on a calculator.