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Math  /  Trigonometry

Question1. Determine each unknown side length to the nearest tenth. a) b)

Studdy Solution

STEP 1

1. Triangle ABC is a non-right triangle.
2. Triangle PQR is a non-right triangle.
3. We will use the Law of Cosines to find the unknown side lengths.
4. All angles are measured in degrees.

STEP 2

1. Use the Law of Cosines to find the unknown side c c in triangle ABC.
2. Use the Law of Cosines to find the unknown side r r in triangle PQR.

STEP 3

For triangle ABC, use the Law of Cosines to find side c c :
The Law of Cosines states: c2=a2+b2−2ab⋅cos⁡(C) c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
Substitute the known values: c2=4.02+7.02−2×4.0×7.0×cos⁡(62∘) c^2 = 4.0^2 + 7.0^2 - 2 \times 4.0 \times 7.0 \times \cos(62^\circ)
Calculate: c2=16+49−56×cos⁡(62∘) c^2 = 16 + 49 - 56 \times \cos(62^\circ)
c2=65−56×cos⁡(62∘) c^2 = 65 - 56 \times \cos(62^\circ)
Calculate cos⁡(62∘) \cos(62^\circ) using a calculator, then compute c2 c^2 .
Find c c by taking the square root and round to the nearest tenth.
STEP_1.1: Calculate cos⁡(62∘) \cos(62^\circ) and complete the computation for c c .
cos⁡(62∘)≈0.4695 \cos(62^\circ) \approx 0.4695
c2=65−56×0.4695 c^2 = 65 - 56 \times 0.4695
c2=65−26.292 c^2 = 65 - 26.292
c2=38.708 c^2 = 38.708
c=38.708 c = \sqrt{38.708}
c≈6.2 c \approx 6.2

STEP 4

For triangle PQR, use the Law of Cosines to find side r r :
The Law of Cosines states: r2=p2+q2−2pq⋅cos⁡(R) r^2 = p^2 + q^2 - 2pq \cdot \cos(R)
Substitute the known values: r2=12.02+9.02−2×12.0×9.0×cos⁡(125∘) r^2 = 12.0^2 + 9.0^2 - 2 \times 12.0 \times 9.0 \times \cos(125^\circ)
Calculate: r2=144+81−216×cos⁡(125∘) r^2 = 144 + 81 - 216 \times \cos(125^\circ)
r2=225−216×cos⁡(125∘) r^2 = 225 - 216 \times \cos(125^\circ)
Calculate cos⁡(125∘) \cos(125^\circ) using a calculator, then compute r2 r^2 .
Find r r by taking the square root and round to the nearest tenth.
STEP_2.1: Calculate cos⁡(125∘) \cos(125^\circ) and complete the computation for r r .
cos⁡(125∘)≈−0.5736 \cos(125^\circ) \approx -0.5736
r2=225−216×(−0.5736) r^2 = 225 - 216 \times (-0.5736)
r2=225+123.4176 r^2 = 225 + 123.4176
r2=348.4176 r^2 = 348.4176
r=348.4176 r = \sqrt{348.4176}
r≈18.7 r \approx 18.7
The unknown side lengths are:
a) c≈6.2 c \approx 6.2
b) r≈18.7 r \approx 18.7

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