Math

QuestionCalculate the value of sin24657cos233+cos24657sin233\sin 246^{\circ} 57^{\prime} \cos 23^{\circ} 3^{\prime} + \cos 246^{\circ} 57^{\prime} \sin 23^{\circ} 3^{\prime}.

Studdy Solution

STEP 1

Assumptions1. The problem requires the use of a calculator to evaluate the expression. . The trigonometric identities of sine and cosine are known.
3. The angles are given in degrees and minutes, where1 degree equals60 minutes.

STEP 2

We need to convert the angles from degrees and minutes to just degrees. We can do this by dividing the minutes by60 and adding the result to the degrees.
24657=246+5760degrees246^{\circ}57^{\prime} =246 + \frac{57}{60} \, degrees23^{\circ}^{\prime} =23 + \frac{}{60} \, degrees

STEP 3

Now, plug in the values and calculate the decimal degree values.
24657=246+5760=246.95degrees246^{\circ}57^{\prime} =246 + \frac{57}{60} =246.95 \, degrees233=23+360=23.05degrees23^{\circ}3^{\prime} =23 + \frac{3}{60} =23.05 \, degrees

STEP 4

Now we can substitute the decimal degree values back into the original expression.
sin246.95cos23.05+cos246.95sin23.05\sin246.95^{\circ} \cos23.05^{\circ}+\cos246.95^{\circ} \sin23.05^{\circ}

STEP 5

We can now use the calculator to evaluate the sine and cosine of the decimal degree values.
sin246.95=0.8912\sin246.95^{\circ} = -0.8912cos23.05=0.9205\cos23.05^{\circ} =0.9205cos246.95=0.453\cos246.95^{\circ} = -0.453sin23.05=0.3910\sin23.05^{\circ} =0.3910

STEP 6

Substitute these values back into the expression.
0.8912×0.9205+0.4536×0.3910-0.8912 \times0.9205 + -0.4536 \times0.3910

STEP 7

Finally, use the calculator to evaluate the expression.
0.8912×0.9205+0.4536×0.3910=0.8200+0.1773=0.9973-0.8912 \times0.9205 + -0.4536 \times0.3910 = -0.8200 + -0.1773 = -0.9973The result of the expression is -0.9973.

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