Right Triangles and Trigonometry Flashcards

In right triangle ABC with the right angle at C, if sin(A) = 2/3, what is the value of sin²(A) + cos²(A)?

Answer: 1

The Pythagorean identity states sin²(θ) + cos²(θ) = 1 for any angle θ.

A right triangle has legs of length 7 and 24. What is the value of sin of the angle opposite the leg of length 7?

Answer: 7/25

The hypotenuse is √(7² + 24²) = √(49 + 576) = √625 = 25, so sin = opposite/hypotenuse = 7/25.

In a 30-60-90 right triangle, if the hypotenuse has length 8, what is the length of the side opposite the 60° angle?

Answer: 4√3

In a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2, so the side opposite 60° = (8/2)·√3 = 4√3.

Right triangle PQR has the right angle at R. If cos(P) = 5/13, what is tan(P)?

Answer: 12/5

If cos(P) = 5/13, then the adjacent leg is 5 and hypotenuse is 13, so the opposite leg is √(169 − 25) = 12, giving tan(P) = 12/5.

A 10-foot ladder leans against a vertical wall. The base of the ladder is 6 feet from the wall. At what angle, to the nearest degree, does the ladder make with the ground?

Answer: 53°

cos(θ) = 6/10 = 0.6, so θ = cos⁻¹(0.6) ≈ 53°.

In right triangle ABC with right angle at C, AB = 10 and AC = 6. What is sin(B)?

Answer: 3/5

AC = 6 is adjacent to B, AB = 10 is the hypotenuse, so BC = √(100 − 36) = 8. sin(B) = opposite/hypotenuse = AC/AB = 6/10 = 3/5.

Two right triangles are similar. The first has legs 6 and 8. The hypotenuse of the second triangle is 25. What is the perimeter of the second triangle?

Answer: 60

The first triangle has hypotenuse √(36+64) = 10. Scale factor = 25/10 = 2.5. Legs of the second = 6×2.5 = 15 and 8×2.5 = 20. Perimeter = 15 + 20 + 25 = 60.