Right Triangles and Trigonometry Flashcards

In right triangle ABC with right angle at C, cos(A) = 4/5. If BC = 6, what is AC?

Answer: 8

cos(A) = AC/AB = 4/5. sin(A) = 3/5, so BC/AB = 3/5. If BC = 6, then AB = 10 and AC = 8.

A right triangle has an acute angle of 45° and an area of 32. What is the length of the hypotenuse?

Answer: 8√2

A 45-45-90 triangle has equal legs; area = (1/2)s² = 32 so s² = 64 and s = 8. Hypotenuse = 8√2.

In a right triangle, one leg is 3 more than the other leg, and the hypotenuse is 15. What is the length of the shorter leg?

Answer: 9

Let the shorter leg be x; then x² + (x+3)² = 225. Expanding: 2x² + 6x + 9 = 225, so 2x² + 6x − 216 = 0, x² + 3x − 108 = 0, (x+12)(x−9) = 0, so x = 9.

In right triangle PQR with right angle at R, sin(P) = 3/5. What is sin(Q)?

Answer: 4/5

Since P and Q are complementary, sin(Q) = cos(P) = √(1 − 9/25) = √(16/25) = 4/5.

A diagonal of a rectangle has length 20 and makes a 30° angle with the longer side. What is the area of the rectangle?

Answer: 100√3

The longer side = 20·cos(30°) = 20·(√3/2) = 10√3. The shorter side = 20·sin(30°) = 20·(1/2) = 10. Area = 10√3 × 10 = 100√3.

In right triangle ABC with right angle at C, AC = 5 and the perimeter is 30. What is BC?

Answer: 12

Let BC = a; hypotenuse AB = √(25 + a²). Perimeter: 5 + a + √(25+a²) = 30, so √(25+a²) = 25 − a. Squaring: 25 + a² = 625 − 50a + a², giving 50a = 600 and a = 12.

Which of the following is equivalent to sin(70°)?

Answer: cos(20°)

sin(70°) = cos(90° − 70°) = cos(20°), using the co-function identity.