Trigonometry Laws of Sines and Cosines 4

Question 1

What is the area of triangle ABC when a = 9, b = 12, and angle C = 50°?

Accepted Answer

Area = ½·9·12·sin 50°

Answer

The area of a triangle given two sides and the included angle is Area = ½·a·b·sin C = ½·9·12·sin 50°.

Question 2

In triangle DEF, d = 7, e = 24, f = 25. Is this triangle a right triangle? What confirms this?

Accepted Answer

Yes, because the Law of Cosines gives cos F = 0

Answer

cos F = (d² + e² − f²) / (2de) = (49 + 576 − 625) / 336 = 0/336 = 0, confirming F = 90°.

Question 3

Using the Law of Sines, if angle A = 120° and angle B = 40°, what is angle C?

Accepted Answer

20°

Answer

Angle C = 180° − 120° − 40° = 20°.

Question 4

Two forces of 50 N and 70 N act on an object at an angle of 130° between them. What is the magnitude of the resultant force?

Accepted Answer

√(2500 + 4900 + 7000·cos 130°)

Answer

The included angle in the parallelogram is 180° − 130° = 50°, but using the triangle with the 130° angle: R² = 50² + 70² − 2(50)(70)cos 130°; since cos 130° is negative, this equals 2500 + 4900 + 7000·|cos 130°|.

Question 5

Which condition guarantees the ambiguous SSA case produces exactly TWO distinct triangles?

Accepted Answer

b·sin A < a < b (with A acute)

Answer

Two triangles result when the given side a is between the altitude (b·sin A) and the other given side b, and angle A is acute.

Trigonometry Practice Test

Trigonometry Practice Test

Trigonometry Laws of Sines and Cosines 4