Right Triangles and Trigonometry Flashcards

In right triangle MNO with the right angle at O, MN = 10 and MO = 6. What is cos(M)?

Answer: 3/5

cos(M) = adjacent/hypotenuse = MO/MN = 6/10 = 3/5.

A tree casts a shadow 24 feet long. A 6-foot person standing nearby casts a shadow 4 feet long. How tall is the tree?

Answer: 36

Similar triangles: tree/24 = 6/4; tree = 6 × 24/4 = 36.

In a 30-60-90 triangle, the hypotenuse is 16. What is the length of the longer leg?

Answer: 8√3

Shorter leg = 16/2 = 8; longer leg = 8√3.

From the top of a 40-foot cliff, the angle of depression to a boat is 30°. How far is the boat from the base of the cliff?

Answer: 40√3

tan(30°) = 40/d; d = 40/tan(30°) = 40/(1/√3) = 40√3.

A right triangle has legs x and x + 7. Its hypotenuse is 13. What is the smaller leg x?

Answer: 5

x² + (x+7)² = 169 → 2x² + 14x + 49 = 169 → 2x² + 14x − 120 = 0 → x² + 7x − 60 = 0 → (x+12)(x−5) = 0; x = 5.

In right triangle ABC with right angle at C, sin(B) = 4/5. If BC = 3, what is AC?

Answer: 4

sin(B) = AC/AB; AB = 3/(cos(B)) — but easier: sides are in ratio 3-4-5; if BC = 3 (adjacent to B), then AB = 5 and AC = 4.

Two right triangles are similar. In the smaller, the legs are 5 and 12. In the larger, the longer leg is 36. What is the hypotenuse of the larger triangle?

Answer: 39

Scale factor = 36/12 = 3; smaller hypotenuse = 13; larger hypotenuse = 13 × 3 = 39.