Right Triangle Trig Quiz for Students and Learners

1) LMN is a right triangle with a hypotenuse of 22 ft. If cos L is 3/4, what is LM?

18.4

16.5

11

7

Cosine equals adjacent divided by hypotenuse. Since cos L is 3/4, LM must be three-fourths of the 22-ft hypotenuse. Multiplying 22 by 3/4 gives 16.5 ft. Other options represent incorrect ratios or incorrect multiplications. The cosine ratio always compares the side touching the angle to the hypotenuse, confirming LM = 16.5 ft.

Explanation

Cosine equals adjacent divided by hypotenuse. Since cos L is 3/4, LM must be three-fourths of the 22-ft hypotenuse. Multiplying 22 by 3/4 gives 16.5 ft. Other options represent incorrect ratios or incorrect multiplications. The cosine ratio always compares the side touching the angle to the hypotenuse, confirming LM = 16.5 ft.

2) A cell tower is 200 ft tall. The angle of elevation is 41°. What is the distance to the tower?

230.1 ft

131.2 ft

150.9 ft

265.0 ft

The distance is the adjacent side in a tangent equation. Using tan(41°) = opposite/adjacent gives tan(41°) = 200/d. Solving for distance gives d = 200 ÷ tan(41°). Using the tangent value for 41 degrees produces approximately 230.1 ft. Other options differ significantly from the actual value and do not match the trig calculation.

Explanation

The distance is the adjacent side in a tangent equation. Using tan(41°) = opposite/adjacent gives tan(41°) = 200/d. Solving for distance gives d = 200 ÷ tan(41°). Using the tangent value for 41 degrees produces approximately 230.1 ft. Other options differ significantly from the actual value and do not match the trig calculation.

3) In a right triangle ABC, the hypotenuse is 17 ft and cos C = 3/5. What is AC?

6

10.2

12

15

Cosine equals adjacent divided by hypotenuse. Given cos C = 3/5, AC must be 3 parts out of a hypotenuse of 5 parts. Scaling this proportion to a hypotenuse of 17, AC = 17 × 3/5 = 10.2 ft. Other values either come from incorrect multiplication or misunderstanding of the ratio.

Explanation

Cosine equals adjacent divided by hypotenuse. Given cos C = 3/5, AC must be 3 parts out of a hypotenuse of 5 parts. Scaling this proportion to a hypotenuse of 17, AC = 17 × 3/5 = 10.2 ft. Other values either come from incorrect multiplication or misunderstanding of the ratio.

4) The perimeter of a square is 36 units. What is the diagonal length?

√162

$9

$12

√200

With perimeter 36, each side is 9 units. The diagonal forms a right triangle with both legs equal to 9. Applying the Pythagorean theorem gives diagonal² = 9² + 9² = 162. The diagonal is √162. Other options do not represent accurate calculations or correspond to the correct triangle geometry.

Explanation

With perimeter 36, each side is 9 units. The diagonal forms a right triangle with both legs equal to 9. Applying the Pythagorean theorem gives diagonal² = 9² + 9² = 162. The diagonal is √162. Other options do not represent accurate calculations or correspond to the correct triangle geometry.

5) A right triangle has an angle of elevation of 32° and the opposite side is 40 ft. What is the adjacent side?

62.5 ft

48.9 ft

64.2 ft

39.4 ft

Using tan(32°) = opposite/adjacent, we have 40/adjacent. Solving gives adjacent = 40 ÷ tan(32°). Computing the value yields about 62.5 ft. All other options either underestimate or overestimate the adjacent side. The tangent ratio ensures precise calculation based on angle and opposite length.

Explanation

Using tan(32°) = opposite/adjacent, we have 40/adjacent. Solving gives adjacent = 40 ÷ tan(32°). Computing the value yields about 62.5 ft. All other options either underestimate or overestimate the adjacent side. The tangent ratio ensures precise calculation based on angle and opposite length.

6) What is tan A in the right triangle ABC?

A/c

B/c

A/b

B/a

For tan A, the tangent ratio equals opposite divided by adjacent. In triangle ABC, side a is opposite angle A and side b is adjacent. Tangent does not involve the hypotenuse, so any option with c is incorrect. Therefore, a/b is the only valid and complete ratio. This directly follows the trigonometric definition of tangent for right triangles.

Explanation

For tan A, the tangent ratio equals opposite divided by adjacent. In triangle ABC, side a is opposite angle A and side b is adjacent. Tangent does not involve the hypotenuse, so any option with c is incorrect. Therefore, a/b is the only valid and complete ratio. This directly follows the trigonometric definition of tangent for right triangles.

7) What is sin A in the right triangle ABC?

B/c

A/c

A/b

B/a

Sine equals opposite divided by hypotenuse. In triangle ABC, a is opposite angle A and c is the hypotenuse. Options with b involve the adjacent side, which does not appear in the sine ratio. Therefore, a/c is the correct representation of sin A.

Explanation

Sine equals opposite divided by hypotenuse. In triangle ABC, a is opposite angle A and c is the hypotenuse. Options with b involve the adjacent side, which does not appear in the sine ratio. Therefore, a/c is the correct representation of sin A.

8) A right triangle has sides written as fractions. Which option represents a valid trigonometric ratio?

Sin A = opposite/hypotenuse

Cos A = hypotenuse/adjacent

Tan A = hypotenuse/opposite

Sin A = adjacent/opposite

A valid ratio must match a standard trig definition. The only correct expression is sin A = opposite/hypotenuse, matching how sine is defined. The other options invert ratios or use incorrect pairings of triangle sides. Trigonometric ratios always follow specific relationships between opposite, adjacent, and hypotenuse sides relative to an angle.

Explanation

A valid ratio must match a standard trig definition. The only correct expression is sin A = opposite/hypotenuse, matching how sine is defined. The other options invert ratios or use incorrect pairings of triangle sides. Trigonometric ratios always follow specific relationships between opposite, adjacent, and hypotenuse sides relative to an angle.

9) What is sin 60?

√3/2

1/2

√2/2

1

In a 30-60-90 triangle, sin 60 equals the ratio of the longer leg to the hypotenuse, which simplifies to √3/2. The 1/2 value corresponds to sin 30, not sin 60. √2/2 corresponds to 45 degrees. Therefore, the only accurate trigonometric value is √3/2.

Explanation

In a 30-60-90 triangle, sin 60 equals the ratio of the longer leg to the hypotenuse, which simplifies to √3/2. The 1/2 value corresponds to sin 30, not sin 60. √2/2 corresponds to 45 degrees. Therefore, the only accurate trigonometric value is √3/2.

10) A ladder rests at a 50° angle with the ground. The base is 10 ft from the house. How long is the ladder?

15.6 ft

6.4 ft

13.1 ft

11.9 ft

The ladder is the hypotenuse, and tan 50° = opposite/adjacent. Here, opposite is the house height, and adjacent is 10 ft. The hypotenuse is solved using the relationship: length = 10 ÷ cos 50°. This produces approximately 15.6 ft. Other options fail to match the trigonometric calculation for the hypotenuse.

Explanation

The ladder is the hypotenuse, and tan 50° = opposite/adjacent. Here, opposite is the house height, and adjacent is 10 ft. The hypotenuse is solved using the relationship: length = 10 ÷ cos 50°. This produces approximately 15.6 ft. Other options fail to match the trigonometric calculation for the hypotenuse.

Right Triangle Trig Quiz On Sine Cosine Tangent