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Evaluate the following trigonometric expressions and round to one decimal place: 1.1 sin 12°, 1.2 (2/3)cos 20°, 1.3 sin 39°/cos 21°, 1.4 2cos 65° + cot 40°, 1.5 2sin 55°cos 55°

Mathematics

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Here are the evaluations, corrected to one decimal place:

Part 1.1· 2 steps

  1. Calculate sin⁡12∘\sin 12^\circ.

    sin⁡12∘≈0.2079\sin 12^\circ \approx 0.2079

  2. Round to one decimal place.

    0.2079≈0.20.2079 \approx 0.2

    The value is 0.2\boxed{0.2}.

Part 1.2· 3 steps

  1. Calculate cos⁡20∘\cos 20^\circ.

    cos⁡20∘≈0.9397\cos 20^\circ \approx 0.9397

  2. Multiply by 23\frac{2}{3}.

    23cos⁡20∘≈23×0.9397≈0.6265\frac{2}{3} \cos 20^\circ \approx \frac{2}{3} \times 0.9397 \approx 0.6265

  3. Round to one decimal place.

    0.6265≈0.60.6265 \approx 0.6

    The value is 0.6\boxed{0.6}.

Part 1.3· 3 steps

  1. Calculate sin⁡39∘\sin 39^\circ and cos⁡21∘\cos 21^\circ.

    sin⁡39∘≈0.6293\sin 39^\circ \approx 0.6293 cos⁡21∘≈0.9336\cos 21^\circ \approx 0.9336

  2. Divide the values.

    sin⁡39∘cos⁡21∘≈0.62930.9336≈0.6740\frac{\sin 39^\circ}{\cos 21^\circ} \approx \frac{0.6293}{0.9336} \approx 0.6740

  3. Round to one decimal place.

    0.6740≈0.70.6740 \approx 0.7

    The value is 0.7\boxed{0.7}.

Part 1.4· 3 steps

  1. Calculate 2cos⁡65∘2 \cos 65^\circ and cot⁡40∘\cot 40^\circ.

    cos⁡65∘≈0.4226\cos 65^\circ \approx 0.4226 2cos⁡65∘≈2×0.4226=0.84522 \cos 65^\circ \approx 2 \times 0.4226 = 0.8452 cot⁡40∘=1tan⁡40∘≈10.8391≈1.1918\cot 40^\circ = \frac{1}{\tan 40^\circ} \approx \frac{1}{0.8391} \approx 1.1918

  2. Add the results.

    2cos⁡65∘+cot⁡40∘≈0.8452+1.1918=2.03702 \cos 65^\circ + \cot 40^\circ \approx 0.8452 + 1.1918 = 2.0370

  3. Round to one decimal place.

    2.0370≈2.02.0370 \approx 2.0

    The value is 2.0\boxed{2.0}.

Part 1.5· 3 steps

  1. Use the double angle identity sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2 \sin \theta \cos \theta.

    2sin⁡55∘cos⁡55∘=sin⁡(2×55∘)=sin⁡110∘2 \sin 55^\circ \cos 55^\circ = \sin(2 \times 55^\circ) = \sin 110^\circ

  2. Calculate sin⁡110∘\sin 110^\circ.

    sin⁡110∘≈0.9397\sin 110^\circ \approx 0.9397

  3. Round to one decimal place.

    0.9397≈0.90.9397 \approx 0.9

    The value is 0.9\boxed{0.9}.

    Given x=39∘x = 39^\circ and y=25∘y = 25^\circ:

Part 2.1· 2 steps

  1. Substitute x=39∘x = 39^\circ and calculate 2sin⁡39∘2 \sin 39^\circ.

    2sin⁡39∘≈2×0.6293=1.25862 \sin 39^\circ \approx 2 \times 0.6293 = 1.2586

  2. Round to one decimal place.

    1.2586≈1.31.2586 \approx 1.3

    The value is 1.3\boxed{1.3}.

Part 2.2· 2 steps

  1. Substitute y=25∘y = 25^\circ and calculate 3cos⁡25∘3 \cos 25^\circ.

    3cos⁡25∘≈3×0.9063=2.71893 \cos 25^\circ \approx 3 \times 0.9063 = 2.7189

  2. Round to one decimal place.

    2.7189≈2.72.7189 \approx 2.7

    The value is 2.7\boxed{2.7}.

Part 2.3· 2 steps

  1. Substitute y=25∘y = 25^\circ and calculate csc⁡225∘\csc^2 25^\circ.

    csc⁡25∘=1sin⁡25∘≈10.4226≈2.3662\csc 25^\circ = \frac{1}{\sin 25^\circ} \approx \frac{1}{0.4226} \approx 2.3662 csc⁡225∘≈(2.3662)2≈5.5989\csc^2 25^\circ \approx (2.3662)^2 \approx 5.5989

  2. Round to one decimal place.

    5.5989≈5.65.5989 \approx 5.6

    The value is 5.6\boxed{5.6}.

Part 2.4· 3 steps

  1. Calculate x+y2\frac{x+y}{2}.

    x+y2=39∘+25∘2=64∘2=32∘\frac{x+y}{2} = \frac{39^\circ + 25^\circ}{2} = \frac{64^\circ}{2} = 32^\circ

  2. Calculate tan⁡32∘\tan 32^\circ.

    tan⁡32∘≈0.6249\tan 32^\circ \approx 0.6249

  3. Round to one decimal place.

    0.6249≈0.60.6249 \approx 0.6

    The value is 0.6\boxed{0.6}.

Part 2.5· 3 steps

  1. Calculate sin⁡x\sin x and sin⁡y\sin y.

    sin⁡39∘≈0.6293\sin 39^\circ \approx 0.6293 sin⁡25∘≈0.4226\sin 25^\circ \approx 0.4226

  2. Add the values.

    sin⁡x+sin⁡y≈0.6293+0.4226=1.0519\sin x + \sin y \approx 0.6293 + 0.4226 = 1.0519

  3. Round to one decimal place.

    1.0519≈1.11.0519 \approx 1.1

    The value is 1.1\boxed{1.1}.

Part 2.6· 3 steps

  1. Calculate x+2yx + 2y.

    x+2y=39∘+2(25∘)=39∘+50∘=89∘x + 2y = 39^\circ + 2(25^\circ) = 39^\circ + 50^\circ = 89^\circ

  2. Calculate 2cos⁡89∘2 \cos 89^\circ.

    2cos⁡89∘≈2×0.0175=0.03502 \cos 89^\circ \approx 2 \times 0.0175 = 0.0350

  3. Round to one decimal place.

    0.0350≈0.00.0350 \approx 0.0

    The value is 0.0\boxed{0.0}.

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