Right Triangles and Trigonometry Flashcards

In right triangle ABC with right angle at C, if the hypotenuse AB = √5 and one leg AC = 1, what is tan(B)?

Answer: 1/2

BC = √(5−1) = 2. Opposite to angle B is AC = 1; adjacent to B is BC = 2. So tan(B) = 1/2.

A right triangle has legs a and b with a > b. The angle opposite b is 30°. Which of the following must be true?

Answer: All of the above

In a 30-60-90 triangle, if b is opposite 30° and a is opposite 60°, then a = b√3. This means b = a/√3 and a/b = √3, so all three are equivalent.

The hypotenuse of a right triangle lies along the x-axis from (0,0) to (10,0). The right angle vertex is at (6, y) with y > 0. What is y?

Answer: 2√6

In a right triangle inscribed in a semicircle, the altitude from the right angle to the hypotenuse = √(product of the two segments). Segments are 6 and 4, so y = √(6 × 4) = √24 = 2√6.

In right triangle MNP with right angle at N, MP = 2 and sin(M) = sin(P). What is the area of the triangle?

Answer: 1

sin(M) = sin(P) implies M = P = 45° (both acute in a right triangle). So it's a 45-45-90 triangle with hypotenuse 2. Each leg = 2/√2 = √2. Area = (1/2)(√2)(√2) = 1.

In right triangle RST with right angle at T, RT = 5 and RS = 13. What is the value of (sin R)(cos R)?

Answer: 60/169

ST = √(169−25) = 12. sin(R) = ST/RS = 12/13; cos(R) = RT/RS = 5/13. Product = (12/13)(5/13) = 60/169.

A right triangle has legs of 9 and 12. What is the sine of the angle between the hypotenuse and the longer leg?

Answer: 3/5

Hypotenuse = 15. The angle between the hypotenuse and the longer leg (12) has opposite side = 9, so sin = 9/15 = 3/5.

In right triangle ABC with right angle at C, the altitude from C to AB has length h. If AC = 6 and BC = 8, what is h?

Answer: Both A and B

AB = 10. Area = (1/2)(6)(8) = 24 = (1/2)(10)(h), so h = 48/10 = 24/5 = 4.8. Both answers A and B are equal.