7. Non-Right Triangles

Practice this topic

A surveyor wishes to find the distance across a river while standing on a small island. If she measures distances of $a=30m$ to one shore, $c=60m$ to the opposite shore, and an angle of $B=100\degree$ between the two shores, find the distance between the two shores.

Use the Law of Cosines to find the angle $C$ , rounded to the nearest tenth.

Solve the triangle: $b=5,c=3,A=100\degree$ .

Solve the triangle: $a=5$ , $b=15$ , $c=13$ .

Solve each triangle. See Examples 2 and 3.

a = 3.0 ft, b = 5.0 ft, c = 6.0 ft

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference. sin 6x sin 2x

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Use the following conditions to solve Exercises 1–4: 4 𝝅 sin α = ----- , ------- < α < 𝝅 5 2 5 𝝅 cos β = ------ , 0 < β < ------ 13 2 Find the exact value of each of the following. cos (α + β)

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Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. In Exercises 1–4, find the exact value of each expression. cos(45° - 30°)

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In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

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In oblique triangle ABC, C = 68°, a = 5, and b = 6. Find c to the nearest tenth.

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In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

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In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression. cos 50° cos 20° + sin 50° sin 20°

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In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle. cos 50° cos 20° + sin 50° sin 20°

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In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression. 5π π 5π π cos ------- cos -------- + sin -------- sin ------- 12 12 12 12

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In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. b. Write the expression as the cosine of an angle. 5π π 5π π cos ------- cos -------- + sin -------- sin ------- 12 12 12 12

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CONCEPT PREVIEW Assume a triangle ABC has standard labeling.

a. Determine whether SAA, ASA, SSA, SAS, or SSS is given.

b. Determine whether the law of sines or the law of cosines should be used to begin solving the triangle.

a, b, and C

Solve each triangle. Approximate values to the nearest tenth.

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Solve each triangle. Approximate values to the nearest tenth.

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Solve each triangle. Approximate values to the nearest tenth.

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Solve each triangle. See Examples 2 and 3.

A = 41.4°, b = 2.78 yd, c = 3.92 yd

Solve each triangle. See Examples 2 and 3.

C = 45.6°, b = 8.94 m, a = 7.23 m

Solve each triangle. See Examples 2 and 3.

a = 9.3 cm, b = 5.7 cm, c = 8.2 cm

Solve each triangle. See Examples 2 and 3.

a = 42.9 m, b = 37.6 m, c = 62.7 m

Solve each triangle. See Examples 2 and 3.

a = 965 ft, b = 876 ft, c = 1240 ft

Solve each triangle. See Examples 2 and 3.

A = 80° 40', b = 143 cm, c = 89.6 cm

Solve each triangle. See Examples 2 and 3.

B = 74.8°, a = 8.92 in., c = 6.43 in.

Solve each triangle. See Examples 2 and 3.

A = 112.8°, b = 6.28 m, c = 12.2 m

CONCEPT PREVIEW Assume a triangle ABC has standard labeling.

a. Determine whether SAA, ASA, SSA, SAS, or SSS is given.

b. Determine whether the law of sines or the law of cosines should be used to begin solving the triangle.

a, B, and C

A plane has an airspeed of 520 mph. The pilot wishes to fly on a bearing of 310°. A wind of 37 mph is blowing from a bearing of 212°. In what direction should the pilot fly, and what will be her ground speed?

Find the force required to keep a 75-lb sled from sliding down an incline that makes an angle of 27° with the horizontal. (Assume there is no friction.)

Open Question

A crane with a counterweight is shown in the figure. Find the horizontal distance between points A and B to the nearest foot.

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A boat travels 15 km per hr in still water. The boat is traveling across a large river, on a bearing of 130°. The current in the river, coming from the west, has a speed of 7 km per hr. Find the resulting speed of the boat and its resulting direction of travel.

Find the exact area of each triangle using the formula 𝓐 = ½ bh, and then verify that Heron's formula gives the same result.

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Find the area of each triangle ABC.

a = 76.3 ft, b = 109 ft, c = 98.8 ft