Worksheet: Right Triangle Trigonometry: Solve for an Angle

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In this worksheet, we will practice finding a missing angle in a right triangle using the appropriate inverse trigonometric function given two side lengths.

Q1:

The height of a ski slope is 16 meters and the length is 20 meters. Find the measure of ∠𝜃 giving the answer to two decimal places.

Q2:

For the given figure, find the measure of angle 𝜃, in degrees, to two decimal places.

Q3:

In the given figure, find the measure of angle 𝜃, in degrees, to two decimal places.

Q4:

Given the following figure, find the lengths of 𝐴𝐵 and 𝐵𝐶 and the measure of ∠𝐴𝐵𝐶 in degrees. Give your answers to two decimal places.

  • A 𝐴 𝐵 = 8 . 7 0 , 𝐵 𝐶 = 1 0 . 5 7 , 𝑚 ∠ 𝐴 𝐵 𝐶 = 3 2 . 0 0 ∘
  • B 𝐴 𝐵 = 5 . 0 9 , 𝐵 𝐶 = 7 . 8 6 , 𝑚 ∠ 𝐴 𝐵 𝐶 = 3 2 . 0 0 ∘
  • C 𝐴 𝐵 = 9 . 8 8 , 𝐵 𝐶 = 1 1 . 5 6 , 𝑚 ∠ 𝐴 𝐵 𝐶 = 3 5 . 0 0 ∘
  • D 𝐴 𝐵 = 9 . 3 2 , 𝐵 𝐶 = 1 1 . 0 8 , 𝑚 ∠ 𝐴 𝐵 𝐶 = 3 7 . 0 0 ∘
  • E 𝐴 𝐵 = 9 . 6 0 , 𝐵 𝐶 = 1 1 . 3 2 , 𝑚 ∠ 𝐴 𝐵 𝐶 = 3 2 . 0 0 ∘

Q5:

Given the following figure, find the lengths of 𝐴𝐶 and 𝐵𝐶 and the measure of ∠𝐵𝐴𝐶 in degrees. Give your answers to two decimal places.

  • A 𝐴 𝐶 = 1 0 . 2 3 , 𝐵 𝐶 = 9 . 4 2 , 𝑚 ∠ 𝐵 𝐴 𝐶 = 6 8 . 0 0 ∘
  • B 𝐴 𝐶 = 1 0 . 6 8 , 𝐵 𝐶 = 9 . 9 0 , 𝑚 ∠ 𝐵 𝐴 𝐶 = 6 9 . 0 0 ∘
  • C 𝐴 𝐶 = 1 0 . 3 4 , 𝐵 𝐶 = 9 . 5 3 , 𝑚 ∠ 𝐵 𝐴 𝐶 = 7 0 . 0 0 ∘
  • D 𝐴 𝐶 = 1 0 . 6 8 , 𝐵 𝐶 = 9 . 9 0 , 𝑚 ∠ 𝐵 𝐴 𝐶 = 6 8 . 0 0 ∘
  • E 𝐴 𝐶 = 1 0 . 5 7 , 𝐵 𝐶 = 9 . 7 8 , 𝑚 ∠ 𝐵 𝐴 𝐶 = 6 8 . 0 0 ∘

Q6:

For the given figure, find the measure of ∠𝐵𝐴𝐶, in degrees, to two decimal places.

Q7:

For the given figure, find the measure of ∠𝐵𝐴𝐶, in degrees, to two decimal places.

Q8:

Find the measure of angle 𝜃, in degrees, to two decimal places.

Q9:

For the given figure, find the measures of ∠𝐴𝐵𝐶 and ∠𝐴𝐶𝐵, in degrees, to two decimal places.

  • A 𝑚 ∠ 𝐴 𝐵 𝐶 = 6 6 . 0 3 ∘ , 𝑚 ∠ 𝐴 𝐶 𝐵 = 2 3 . 9 6 ∘
  • B 𝑚 ∠ 𝐴 𝐵 𝐶 = 6 3 . 6 1 ∘ , 𝑚 ∠ 𝐴 𝐶 𝐵 = 2 6 . 3 9 ∘
  • C 𝑚 ∠ 𝐴 𝐵 𝐶 = 2 6 . 3 9 ∘ , 𝑚 ∠ 𝐴 𝐶 𝐵 = 6 3 . 6 1 ∘
  • D 𝑚 ∠ 𝐴 𝐵 𝐶 = 2 6 . 5 7 ∘ , 𝑚 ∠ 𝐴 𝐶 𝐵 = 6 3 . 4 3 ∘
  • E 𝑚 ∠ 𝐴 𝐵 𝐶 = 6 3 . 4 3 ∘ , 𝑚 ∠ 𝐴 𝐶 𝐵 = 2 6 . 5 7 ∘

Q10:

Given the following figure, find the lengths of 𝐴𝐵 and 𝐵𝐶 and the measure of ∠𝐴𝐶𝐵 in degrees. Give your answers to two decimal places.

  • A 𝐴 𝐵 = 1 0 . 8 7 , 𝐵 𝐶 = 1 2 . 4 2 , 𝑚 ∠ 𝐴 𝐶 𝐵 = 5 7 . 0 0 ∘
  • B 𝐴 𝐵 = 9 . 2 4 , 𝐵 𝐶 = 1 1 . 0 2 , 𝑚 ∠ 𝐴 𝐶 𝐵 = 5 6 . 0 0 ∘
  • C 𝐴 𝐵 = 4 . 8 2 , 𝐵 𝐶 = 1 0 . 4 2 , 𝑚 ∠ 𝐴 𝐶 𝐵 = 5 7 . 0 0 ∘
  • D 𝐴 𝐵 = 1 0 . 8 7 , 𝐵 𝐶 = 1 2 . 4 2 , 𝑚 ∠ 𝐴 𝐶 𝐵 = 6 0 . 0 0 ∘
  • E 𝐴 𝐵 = 9 . 2 4 , 𝐵 𝐶 = 1 1 . 0 2 , 𝑚 ∠ 𝐴 𝐶 𝐵 = 5 7 . 0 0 ∘

Q11:

Given the following figure, find the lengths of 𝐴𝐵 and 𝐴𝐶 and the measure of 𝑚∠𝐵𝐶𝐴 in degrees. Give your answers to two decimal places.

  • A 𝐴 𝐵 = 7 . 0 3 , 𝐴 𝐶 = 5 . 5 4 , 𝑚 ∠ 𝐵 𝐶 𝐴 = 5 3 . 0 0 ∘
  • B 𝐴 𝐵 = 7 . 0 9 , 𝐴 𝐶 = 5 . 5 4 , 𝑚 ∠ 𝐵 𝐶 𝐴 = 5 2 . 0 0 ∘
  • C 𝐴 𝐵 = 7 . 0 3 , 𝐴 𝐶 = 5 . 5 4 , 𝑚 ∠ 𝐵 𝐶 𝐴 = 5 2 . 0 0 ∘
  • D 𝐴 𝐵 = 7 . 0 9 , 𝐴 𝐶 = 5 . 5 4 , 𝑚 ∠ 𝐵 𝐶 𝐴 = 5 3 . 0 0 ∘
  • E 𝐴 𝐵 = 6 . 5 8 , 𝐴 𝐶 = 6 . 1 4 , 𝑚 ∠ 𝐵 𝐶 𝐴 = 5 2 . 0 0 ∘

Q12:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝐵𝐶=25cm and 𝑚∠𝐴=58∘. Find the lengths of 𝐴𝐶 and 𝐴𝐵 and the measure of ∠𝐶.

  • A 𝐴 𝐶 = 2 9 . 4 8 c m , 𝐴 𝐵 = 1 5 . 6 2 c m , 𝑚 ∠ 𝐶 = 4 2 ∘
  • B 𝐴 𝐶 = 1 5 . 6 2 c m , 𝐴 𝐵 = 2 9 . 4 8 c m , 𝑚 ∠ 𝐶 = 4 2 ∘
  • C 𝐴 𝐶 = 1 5 . 6 2 c m , 𝐴 𝐵 = 2 9 . 4 8 c m , 𝑚 ∠ 𝐶 = 3 2 ∘
  • D 𝐴 𝐶 = 2 9 . 4 8 c m , 𝐴 𝐵 = 1 5 . 6 2 c m , 𝑚 ∠ 𝐶 = 3 2 ∘

Q13:

For the given figure, find angle 𝜃, in degrees, to two decimal places.

Q14:

For the given figure, find the measures of 𝑚∠𝐴𝐶𝐵 and 𝑚∠𝐵𝐴𝐶, in degrees, to two decimal places.

  • A 𝑚 ∠ 𝐴 𝐶 𝐵 = 5 3 . 1 3 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 3 6 . 8 7 ∘
  • B 𝑚 ∠ 𝐴 𝐶 𝐵 = 3 8 . 6 6 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 5 1 . 3 4 ∘
  • C 𝑚 ∠ 𝐴 𝐶 𝐵 = 5 1 . 3 4 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 3 8 . 6 6 ∘
  • D 𝑚 ∠ 𝐴 𝐶 𝐵 = 3 6 . 8 7 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 5 3 . 1 3 ∘
  • E 𝑚 ∠ 𝐴 𝐶 𝐵 = 3 7 . 9 9 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 5 2 . 0 1 ∘

Q15:

𝐴 𝐵 𝐶 is a right triangle at 𝐵, where 𝐵𝐶=10cm and 𝐴𝐶=18cm. Find the length 𝐴𝐵, giving the answer to the nearest centimeter, and the measure of angles 𝐴 and 𝐶, giving the answer to the nearest degree.

  • A 𝐴 𝐵 = 1 5 c m , 𝑚 ∠ 𝐴 = 4 3 ∘ , 𝑚 ∠ 𝐶 = 4 7 ∘
  • B 𝐴 𝐵 = 1 5 c m , 𝑚 ∠ 𝐴 = 3 4 ∘ , 𝑚 ∠ 𝐶 = 5 6 ∘
  • C 𝐴 𝐵 = 1 5 c m , 𝑚 ∠ 𝐴 = 3 5 ∘ , 𝑚 ∠ 𝐶 = 5 5 ∘
  • D 𝐴 𝐵 = 1 5 c m , 𝑚 ∠ 𝐴 = 4 2 ∘ , 𝑚 ∠ 𝐶 = 4 8 ∘

Q16:

James leans a 5 m ladder against a wall that is perpendicular to the ground. He positions the ladder such that the angle between the base of the ladder and the ground is 72∘. Determine the height, ℎ, at which the top of the ladder touches the wall, the angle between the top of the ladder and the wall, 𝜃, and the distance, 𝑑, between the bottom of the ladder and the base of the wall. If necessary, round your answers to two decimal places.

  • A ℎ = 4 . 7 6 m, 𝜃 = 5 7 ∘ , 𝑑 = 1 . 5 5 m
  • B ℎ = 5 . 2 6 m, 𝜃 = 1 8 ∘ , 𝑑 = 1 . 5 5 m
  • C ℎ = 1 . 5 5 m, 𝜃 = 5 7 ∘ , 𝑑 = 4 . 7 6 m
  • D ℎ = 1 . 5 5 m, 𝜃 = 1 8 ∘ , 𝑑 = 4 . 7 6 m
  • E ℎ = 4 . 7 6 m, 𝜃 = 1 8 ∘ , 𝑑 = 1 . 5 5 m

Q17:

The height of a ski slope is 4 meters and the length is 5 meters. Find the measure of ∠𝜃 giving the answer to two decimal places.

Q18:

The height of a ski slope is 3 meters and the length is 5 meters. Find the measure of ∠𝜃 giving the answer to two decimal places.

Q19:

𝐴 𝐵 𝐶 is a right triangle at 𝐵, where 𝐵𝐶=13.8cm and 𝐴𝐶=19cm. Find the length 𝐴𝐵, giving the answer to the nearest centimeter, and the measure of angles 𝐴 and 𝐶, giving the answer to the nearest degree.

  • A 𝐴 𝐵 = 1 3 c m , 𝑚 ∠ 𝐴 = 3 7 ∘ , 𝑚 ∠ 𝐶 = 5 3 ∘
  • B 𝐴 𝐵 = 1 3 c m , 𝑚 ∠ 𝐴 = 4 7 ∘ , 𝑚 ∠ 𝐶 = 4 3 ∘
  • C 𝐴 𝐵 = 1 3 c m , 𝑚 ∠ 𝐴 = 4 8 ∘ , 𝑚 ∠ 𝐶 = 4 2 ∘
  • D 𝐴 𝐵 = 1 3 c m , 𝑚 ∠ 𝐴 = 3 6 ∘ , 𝑚 ∠ 𝐶 = 5 4 ∘

Q20:

Given the following figure, find the measures of angles ∠𝐴𝐶𝐵 and ∠𝐵𝐴𝐶 and the length of 𝐴𝐶. Give your answers to two decimal places.

  • A 𝑚 ∠ 𝐴 𝐶 𝐵 = 4 1 . 6 3 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 4 8 . 3 7 ∘ , 𝐴 𝐶 = 9 . 4 3
  • B 𝑚 ∠ 𝐴 𝐶 𝐵 = 4 1 . 6 3 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 4 8 . 3 7 ∘ , 𝐴 𝐶 = 1 2 . 0 4
  • C 𝑚 ∠ 𝐴 𝐶 𝐵 = 4 8 . 3 7 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 4 1 . 6 3 ∘ , 𝐴 𝐶 = 9 . 4 3
  • D 𝑚 ∠ 𝐴 𝐶 𝐵 = 4 8 . 3 7 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 4 1 . 6 3 ∘ , 𝐴 𝐶 = 1 1 . 2 2
  • E 𝑚 ∠ 𝐴 𝐶 𝐵 = 6 2 . 7 3 ∘ , 𝑚 ∠ 𝐵 𝐴 𝐶 = 2 7 . 2 7 ∘ , 𝐴 𝐶 = 1 0 . 6 3

Q21:

Find the measure of ∠𝜃 giving the answer to the nearest second.

  • A 5 7 5 ′ 4 1 ′ ′ ∘
  • B 4 9 4 0 ′ 4 7 ′ ′ ∘
  • C 4 0 1 9 ′ 1 3 ′ ′ ∘
  • D 3 2 5 4 ′ 1 9 ′ ′ ∘

Q22:

A palm tree 10.6 metres tall is observed from a point 12 metres away on the same horizontal plane as the base of the tree. Find the angle of elevation to the top of the palm tree giving the answer to the nearest minute.

  • A 2 7 5 7 ′ ∘
  • B 6 2 3 ′ ∘
  • C 4 1 2 7 ′ ∘
  • D 4 8 3 3 ′ ∘

Q23:

Find the value of ∠𝐴𝐶𝐵 giving the answer to the nearest second.

  • A 4 4 2 4 ′ 5 5 ′ ′ ∘
  • B 3 5 3 2 ′ 1 6 ′ ′ ∘
  • C 4 5 3 5 ′ 5 ′ ′ ∘
  • D 5 4 2 7 ′ 4 4 ′ ′ ∘

Q24:

Find the measure of ∠𝐴𝐶𝐵 given 𝐴𝐵𝐶𝐷 is a rectangle where 𝐴𝐵=10cm and 𝐴𝐶=26cm. Give the answer to the nearest second.

  • A 6 7 2 2 ′ 4 8 ′ ′ ∘
  • B 2 4 3 7 ′ 2 8 ′ ′ ∘
  • C 2 1 2 ′ 1 5 ′ ′ ∘
  • D 2 2 3 7 ′ 1 2 ′ ′ ∘

Q25:

Find the measure of ∠𝜃 giving the answer to the nearest second.

  • A 2 1 1 5 ′ 2 ′ ′ ∘
  • B 6 8 4 4 ′ 5 8 ′ ′ ∘
  • C 6 7 6 ′ 5 3 ′ ′ ∘
  • D 2 2 5 3 ′ 7 ′ ′ ∘

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