Worksheet: Right Triangle Trigonometry: Solve for a Side

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 𝐵𝐶 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 𝐴𝐶 and 𝐵𝐶 and the measure of 𝐴𝐵𝐶 in degrees. Give your answers to two decimal places.

  • A𝐴𝐶=7.82, 𝐵𝐶=8.37, 𝑚𝐴𝐵𝐶=69.00
  • B𝐴𝐶=7.22, 𝐵𝐶=7.82, 𝑚𝐴𝐵𝐶=69.00
  • C𝐴𝐶=7.32, 𝐵𝐶=7.91, 𝑚𝐴𝐵𝐶=72.00
  • D𝐴𝐶=7.95, 𝐵𝐶=8.19, 𝑚𝐴𝐵𝐶=60.00
  • E𝐴𝐶=7.62, 𝐵𝐶=8.19, 𝑚𝐴𝐵𝐶=70.00

Q5:

A 23 ft ladder leans against a building such that the angle between the ground and the ladder is 80. 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 60 to the horizontal. Find the length of the string accurate to one decimal place.

Q7:

Find the length of 𝐴𝐶, given that 𝐴𝐵=19cm, 𝐷𝐸=11cm, and 𝐶𝐸=21cm.

  • A6098 cm
  • B39911 cm
  • C23129 cm
  • D11399 cm
  • E31921 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 29. Find the distance between the point and the observer giving the answer to the nearest meter.

Q9:

𝐴𝐵𝐶 is an isosceles triangle where 𝐴𝐵=𝐴𝐶=9cm, 𝐴𝐷𝐵𝐶 and 𝑚𝐶=34. Find the length of 𝐵𝐶 giving the answer to one decimal place.

Q10:

Find the length of 𝐴𝐶 given 𝐴𝐵𝐶 is a right-angled triangle at 𝐵 where sin𝐶=916 and 𝐴𝐵=18cm.

Q11:

A kite has a string of length 75 meters. The angle the string makes with the horizontal ground is 53. Find the height of the kite from the ground giving the answer to two decimal places.

Q12:

𝐴𝐵𝐶 is a triangle where 𝐴𝐶=34cm, 𝐵𝐶=40cm, and 𝑚𝐶=46. Point 𝐷 lies on 𝐶𝐵 where 𝐴𝐷𝐵𝐶. Find the height of the triangle drawn from point 𝐵 to 𝐴𝐶 giving the answer to two decimal places.

Q13:

𝐴𝐵𝐶𝐷 is a rectangle where 𝐴𝐶=49.6cm and 𝑚𝐴𝐶𝐵=2654. Find the lengths of 𝐴𝐵 and 𝐵𝐶 giving the answer to three decimal places.

  • A𝐴𝐵=22.441cm, 𝐵𝐶=44.233cm
  • B𝐴𝐵=44.233cm, 𝐵𝐶=22.441cm
  • C𝐴𝐵=44.233cm, 𝐵𝐶=25.164cm
  • D𝐴𝐵=25.164cm, 𝐵𝐶=44.233cm

Q14:

Find the length of 𝐵𝐶 giving the answer to two decimal places.

Q15:

In the given figure, 𝐴𝐵=5, 𝐵𝐶=𝑎, and 𝐴𝐶=10. Unless otherwise stated, give all solutions to the following questions to four decimal places.

Work out the length of 𝐴𝐷 and 𝐷𝐶.

  • A2.8512, 7.1488
  • B3.2184, 6.7816
  • C1.7101, 8.2899
  • D3.2899, 6.7101
  • E4.6985, 5.3015

Using the Pythagorean theorem, or otherwise, calculate (𝐵𝐷).

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

Q16:

Given the following figure, find 𝑚𝐵𝐴𝐶 and 𝑚𝐴𝐶𝐵 and the length of 𝐴𝐶. Give your answers to two decimal places.

  • A𝑚𝐵𝐴𝐶=59.04, 𝑚𝐴𝐶𝐵=30.96, 𝐴𝐶=8.29
  • B𝑚𝐵𝐴𝐶=36.87, 𝑚𝐴𝐶𝐵=53.13, 𝐴𝐶=6.78
  • C𝑚𝐵𝐴𝐶=59.04, 𝑚𝐴𝐶𝐵=30.96, 𝐴𝐶=5.83
  • D𝑚𝐵𝐴𝐶=36.35, 𝑚𝐴𝐶𝐵=53.65, 𝐴𝐶=4.52
  • E𝑚𝐵𝐴𝐶=53.13, 𝑚𝐴𝐶𝐵=36.87, 𝐴𝐶=5.38

Q17:

In the given figure, given that 𝐴𝐶=10 and 𝑚𝐴𝐶𝐵=50, find the lengths of 𝐴𝐵 and 𝐵𝐶 and the measure of 𝐴𝐵𝐶 in degrees. Give your answers to two decimal places.

  • A𝐴𝐵=7.66, 𝐵𝐶=12.60, 𝑚𝐴𝐵𝐶=41.00
  • B𝐴𝐵=6.42, 𝐵𝐶=11.88, 𝑚𝐴𝐵𝐶=40.00
  • C𝐴𝐵=8.39, 𝐵𝐶=13.05, 𝑚𝐴𝐵𝐶=38.00
  • D𝐴𝐵=11.92, 𝐵𝐶=15.56, 𝑚𝐴𝐵𝐶=40.00
  • E𝐴𝐵=11.92, 𝐵𝐶=15.56, 𝑚𝐴𝐵𝐶=42.00

Q18:

Find 𝑥 in the given figure. Give your answer to two decimal places.

Q19:

Find 𝑚𝐵 given 𝐴𝐵𝐶 is a triangle where 𝑚𝐴=111 and sincos𝐶=𝐶.

Q20:

Find 𝑚𝐴 given 𝐴𝐵𝐶 is a right-angled triangle at 𝐵 where 2𝐴𝐵=𝐴𝐶.

Q21:

Find the values of 𝑥 and 𝑦 giving the answer to three decimal places.

  • A𝑥=8.999cm, 𝑦=16.643cm
  • B𝑥=10.725cm, 𝑦=8.999cm
  • C𝑥=8.999cm, 𝑦=10.725cm
  • D𝑥=16.643cm, 𝑦=8.999cm

Q22:

Find the values of 𝑥 and 𝑦 giving the answer to three decimal places.

  • A𝑥=38.285cm, 𝑦=26.110cm
  • B𝑥=47.432cm, 𝑦=38.285cm
  • C𝑥=26.110cm, 𝑦=38.285cm
  • D𝑥=38.285cm, 𝑦=47.432cm

Q23:

𝑋𝑌𝑍 is a right-angled triangle at 𝑌, where 𝑋𝑌=16.5cm, 𝑌𝑍=28cm, and 𝑋𝑍=32.5cm. Find the measure of 𝑍 giving the answer to the nearest second.

  • A592923
  • B26550
  • C404446
  • D303037

Q24:

Find the value of sincos𝐵+𝐶, given that 𝐴𝐵𝐶 is a triangle, where 𝐴𝐵=𝐴𝐶=41cm, 𝐵𝐶=18cm, and 𝐴𝐷 is drawn perpendicular to 𝐵𝐶 intersecting at 𝐷.

  • A369982
  • B4941
  • C1,180369
  • D4149

Q25:

Find the value of 625𝐴𝐴sincos given 𝐴𝐵𝐶 is a right-angled triangle at 𝐵 where 7𝐴24=0tan.

  • A168
  • B600
  • C175
  • D625

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