Worksheet: Right Triangle Trigonometry

In this worksheet, we will practice finding the value of a missing side length in a right triangle by choosing the appropriate trigonometric ratio for a given angle.

Q1:

Find the length of 𝐵 𝐶 giving the answer to two decimal places.

Q2:

Find 𝑥 to two decimal places.

Q3:

Find 𝑥 to two decimal places.

Q4:

Given the following figure, find the lengths of 𝐴 𝐶 and 𝐵 𝐶 and the measure of 𝐴 𝐵 𝐶 in degrees. Give your answers to two decimal places.

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

Q5:

A 23 ft ladder leans against a building such that the angle between the ground and the ladder is 8 0 . How high does the ladder reach up the side of the building? Give your answer to two decimal places.

Q6:

A kite, which is at a perpendicular height of 44 m, is attached to a string inclined at 6 0 to the horizontal. Find the length of the string accurate to one decimal place.

Q7:

Find the length of 𝐴 𝐶 , given that 𝐴 𝐵 = 1 9 c m , 𝐷 𝐸 = 1 1 c m , and 𝐶 𝐸 = 2 1 c m .

  • A 3 9 9 1 1 cm
  • B 1 1 3 9 9 cm
  • C 6 0 9 8 cm
  • D 3 1 9 2 1 cm
  • E 2 3 1 2 9 cm

Q8:

A person observes a point on the ground from the top of a hill that is 1.56 km high. The angle of depression is 2 9 . Find the distance between the point and the observer giving the answer to the nearest metre.

Q9:

𝐴 𝐵 𝐶 is an isosceles triangle where 𝐴 𝐵 = 𝐴 𝐶 = 9 c m , 𝐴 𝐷 𝐵 𝐶 and 𝑚 𝐶 = 3 4 . Find the length of 𝐵 𝐶 giving the answer to one decimal place.

Q10:

In the given figure, find 𝑥 . Give your answer to two decimal places.

Q11:

Find the length of 𝐴 𝐶 given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where s i n 𝐶 = 9 1 6 and 𝐴 𝐵 = 1 8 c m .

Q12:

A builder leans a 6-foot plank of wood against a vertical wall. If one end of the plank is 1.5 feet from the bottom of the wall, what is the angle between the plank and the wall?

Q13:

A kite has a string of length 75 meters. The angle the string makes with the horizontal ground is 5 3 . Find the height of the kite from the ground giving the answer to two decimal places.

Q14:

The angle of elevation between an aeroplane 10,000 m high and a person on a point on the ground is 3 8 6 . Find the distance between the point and the aeroplane giving the answer to the nearest metre.

Q15:

𝐴 𝐵 𝐶 is a triangle where 𝐴 𝐶 = 3 4 c m , 𝐵 𝐶 = 4 0 c m , and 𝑚 𝐶 = 4 6 . Point 𝐷 lies on 𝐶 𝐵 where 𝐴 𝐷 𝐵 𝐶 . Find the height of the triangle drawn from point 𝐵 to 𝐴 𝐶 giving the answer to two decimal places.

Q16:

𝐴 𝐵 𝐶 𝐷 is a rectangle where 𝐴 𝐶 = 4 9 . 6 c m and 𝑚 𝐴 𝐶 𝐵 = 2 6 5 4 . Find the lengths of 𝐴 𝐵 and 𝐵 𝐶 giving the answer to three decimal places.

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

Q17:

Find the length of 𝐵 𝐶 giving the answer to two decimal places.

Q18:

In the given figure, 𝐴 𝐵 = 5 , 𝐵 𝐶 = 𝑎 , and 𝐴 𝐶 = 1 0 . Unless otherwise stated, give all solutions to the following questions to four decimal places.

Work out the length of 𝐴 𝐷 and 𝐷 𝐶 .

  • A3.2184, 6.7816
  • B4.6985, 5.3015
  • C1.7101, 8.2899
  • D2.8512, 7.1488
  • E3.2899, 6.7101

Using the Pythagorean theorem, or otherwise, calculate ( 𝐵 𝐷 ) .

Work out 𝑎 . Give your answer to two decimal places.

Q19:

Given the following figure, find 𝑚 𝐵 𝐴 𝐶 and 𝑚 𝐴 𝐶 𝐵 and the length of 𝐴 𝐶 . Give your answers to two decimal places.

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

Q20:

In the given figure, given that 𝐴 𝐶 = 1 0 and 𝑚 𝐴 𝐶 𝐵 = 5 0 , find the lengths of 𝐴 𝐵 and 𝐵 𝐶 and the measure of 𝐴 𝐵 𝐶 in degrees. Give your answers to two decimal places.

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

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