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Determine the unknown angles rounded off to one decimal place:

Mathematics

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Answers

4 parts
a

53.1∘53.1^\circ

b

36.9∘36.9^\circ

c

19.9∘19.9^\circ

d

44.8∘44.8^\circ

Step-by-step solution

17 steps

Part a· 4 steps

  1. Identify the known sides relative to angle α\alpha.

    The side opposite to α\alpha is AC=8AC = 8.
    The side adjacent to α\alpha is AB=6AB = 6.

  2. Use the tangent function to find α\alpha.

    tan⁡(α)=oppositeadjacent=86\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{6}

  3. Calculate α\alpha.

    α=arctan⁡(86)\alpha = \arctan\left(\frac{8}{6}\right) α≈53.1301∘\alpha \approx 53.1301^\circ

  4. Round to one decimal place.

    α≈53.1∘\alpha \approx \boxed{53.1^\circ}

Part b

36.9∘36.9^\circ

  1. Identify the known sides relative to angle β\beta.

    The side opposite to β\beta is 3.
    The hypotenuse is 5.

  2. Use the sine function to find β\beta.

    sin⁡(β)=oppositehypotenuse=35\sin(\beta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}

  3. Calculate β\beta.

    β=arcsin⁡(35)\beta = \arcsin\left(\frac{3}{5}\right) β≈36.8699∘\beta \approx 36.8699^\circ

  4. Round to one decimal place.

    β≈36.9∘\beta \approx \boxed{36.9^\circ}

Part c

19.9∘19.9^\circ

  1. Identify the known sides relative to angle (2θ+3∘)(2\theta + 3^\circ).

    The side opposite to (2θ+3∘)(2\theta + 3^\circ) is 12.
    The side adjacent to (2θ+3∘)(2\theta + 3^\circ) is 13.

  2. Use the tangent function to find the angle (2θ+3∘)(2\theta + 3^\circ).

    tan⁡(2θ+3∘)=oppositeadjacent=1213\tan(2\theta + 3^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{13}

  3. Calculate the value of the angle.

    2θ+3∘=arctan⁡(1213)2\theta + 3^\circ = \arctan\left(\frac{12}{13}\right) 2θ+3∘≈42.7086∘2\theta + 3^\circ \approx 42.7086^\circ

  4. Solve for θ\theta.

    2θ=42.7086∘−3∘2\theta = 42.7086^\circ - 3^\circ 2θ=39.7086∘2\theta = 39.7086^\circ θ=39.7086∘2\theta = \frac{39.7086^\circ}{2} θ≈19.8543∘\theta \approx 19.8543^\circ

  5. Round to one decimal place.

    θ≈19.9∘\theta \approx \boxed{19.9^\circ}

Part d

44.8∘44.8^\circ

  1. Identify the known sides relative to angle β\beta.

    The side adjacent to β\beta is 8.3.
    The hypotenuse is 11.7.

  2. Use the cosine function to find β\beta.

    cos⁡(β)=adjacenthypotenuse=8.311.7\cos(\beta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8.3}{11.7}

  3. Calculate β\beta.

    β=arccos⁡(8.311.7)\beta = \arccos\left(\frac{8.3}{11.7}\right) β≈44.7955∘\beta \approx 44.7955^\circ

  4. Round to one decimal place.

    β≈44.8∘\beta \approx \boxed{44.8^\circ}
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