Worksheet: The Tangent Ratio

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In this worksheet, we will practice finding an angle in a right triangle using the tangent ratio.

Q1:

Find the values of 𝛼 and 𝛽 giving your answer to the nearest second.

  • A 𝛼 = 5 4 2 0 ′ 4 1 ′ ′ ∘ , 𝛽 = 3 5 3 9 ′ 1 9 ′ ′ ∘
  • B 𝛼 = 5 4 2 0 ′ 4 1 ′ ′ ∘ , 𝛽 = 3 0 1 4 ′ 1 8 ′ ′ ∘
  • C 𝛼 = 3 0 1 4 ′ 1 8 ′ ′ ∘ , 𝛽 = 5 4 2 0 ′ 4 1 ′ ′ ∘
  • D 𝛼 = 3 5 3 9 ′ 1 9 ′ ′ ∘ , 𝛽 = 5 4 2 0 ′ 4 1 ′ ′ ∘

Q2:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵, where 𝐴𝐵=45cm and 𝐵𝐶=68cm. Find the measure of ∠𝐶 giving the answer to the nearest second.

  • A 3 3 2 9 ′ 4 3 ′ ′ ∘
  • B 4 1 2 6 ′ 5 ′ ′ ∘
  • C 5 6 3 0 ′ 1 7 ′ ′ ∘
  • D 4 8 3 3 ′ 5 5 ′ ′ ∘

Q3:

𝐴 𝐵 𝐶 𝐷 is a rhombus where 𝐴𝐶=24cm and 𝐵𝐷=32cm. Find 𝑚∠𝐵𝐴𝐶 giving the answer to the nearest second.

  • A 3 8 3 9 ′ 3 5 ′ ′ ∘
  • B 5 9 2 ′ 1 0 ′ ′ ∘
  • C 3 6 5 2 ′ 1 2 ′ ′ ∘
  • D 5 3 7 ′ 4 8 ′ ′ ∘

Q4:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝐴𝐵=19cm and 𝑚∠𝐶=54∘. Find the length of 𝐵𝐶 giving the answer to two decimal places.

Q5:

𝐴 𝐵 𝐶 is a triangle where 𝐴𝐷⟂𝐵𝐶, 𝑚∠𝐵=5116′18′′∘, 𝐵𝐶=20cm, and 𝐴𝐷=8cm. Find 𝑚∠𝐶 giving the answer to the nearest second.

  • A 5 3 5 5 ′ 1 2 ′ ′ ∘
  • B 3 0 2 9 ′ 4 0 ′ ′ ∘
  • C 1 6 5 0 ′ 5 6 ′ ′ ∘
  • D 3 6 4 ′ 4 8 ′ ′ ∘

Q6:

The following figure represents a line segment joining two points 𝐴(2,0) and 𝐵(3,8). Find the measure of the angle 𝜃 included between 𝐴𝐵 and the 𝑥-axis giving the answer to the nearest second.

  • A 7 5 5 7 ′ 5 0 ′ ′ ∘
  • B 8 2 5 2 ′ 3 0 ′ ′ ∘
  • C 6 9 2 6 ′ 3 8 ′ ′ ∘
  • D 2 6 3 3 ′ 5 4 ′ ′ ∘

Q7:

A flagpole 5.9 meters tall casts a 2.8-meter shadow. Find the angle of inclination of the sun giving the answer to the nearest minute.

  • A 2 8 2 0 ∘
  • B 6 1 4 0 ∘
  • C 6 4 3 7 ∘
  • D 2 5 2 3 ∘

Q8:

Find the value of 𝐾, given 𝐴𝐵𝐶 is an equilateral triangle, where point 𝐷 lies on 𝐴𝐵, 𝐴𝐷=5cm, 𝐷𝐵=12cm, and 𝐾𝑋=√3tan.

  • A 1 2 1 7
  • B 1 7 1 2
  • C 6 1 1
  • D 1 1 6

Q9:

Find 𝑥 to two decimal places.

Q10:

In the given figure, 𝑚∠𝐵𝐴𝐶=90∘ and 𝐴𝐷⊥𝐵𝐶. What is𝐴𝐶𝜃tan?

  • A 𝐵 𝐶
  • B 𝐴 𝐷
  • C 𝐶 𝐷
  • D 𝐷 𝐵
  • E 𝐴 𝐵

Q11:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵, where 𝐴𝐵=64cm. Point 𝐷 lies on 𝐵𝐶 and point 𝐸 lies on 𝐴𝐶 where 𝐸𝐷∥𝐴𝐵. Find the length of 𝐷𝐵 given 𝐶𝐷=12cm and 𝐸𝐷=35cm. Give the answer to two decimal places.

Q12:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝑚∠𝐶=3412′∘ and 𝐵𝐶=25cm. Find the length of 𝐴𝐵 giving the answer to two decimal places.

Q13:

Find the exact value of tan𝐶.

  • A 4 9
  • B 9 4
  • C 5 4
  • D 4 5

Q14:

The dimensions of a rectangular garden is 13 metres by 12 metres. Triangle 𝐴𝐸𝐶 represents the lawn where 𝐸𝐶=10m. Find the measure of ∠𝐸𝐴𝐵 giving the answer to the nearest second.

  • A 4 7 1 7 ′ 2 6 ′ ′ ∘
  • B 3 9 4 8 ′ 2 0 ′ ′ ∘
  • C 1 4 2 ′ 1 0 ′ ′ ∘
  • D 7 5 5 7 ′ 5 0 ′ ′ ∘

Q15:

𝑋 𝑌 𝑍 is a triangle where 𝐿 lies on 𝑋𝑍 such that 𝑌𝐿⟂𝑋𝑍, tantan𝑍=𝑋𝑌𝐿=2021 and 𝑍𝐿=26cm. Find the area of 𝑋𝑌𝑍 giving the answer to two decimal places.

Q16:

Find the length of 𝐷𝐶 given 𝐴𝐵𝐶𝐷 is a quadrilateral where 𝐴𝐷=6cm and 𝐴𝐵=8cm.

  • A 4 4 5 cm
  • B 1 3 1 3 cm
  • C10 cm
  • D 7 1 2 cm

Q17:

𝐴 𝐵 𝐶 is a triangle where 𝐴𝐷⟂𝐵𝐶, 𝐴𝐷=16cm, 𝑚∠𝐵=44∘, and 𝑚∠𝐶=48∘. Find, to the nearest centimetre, the length of 𝐵𝐶.

Q18:

In the given figure, 𝑚∠𝐵𝐴𝐶=90∘ where 𝐴𝐷⊥𝐵𝐶. What is 𝐴𝐷𝜃tan?

  • A 𝐴 𝐵
  • B 𝐴 𝐷
  • C 𝐴 𝐶
  • D 𝐵 𝐶
  • E 𝐵 𝐷

Q19:

Find the value of tantan𝑥+𝑦 given 𝐴𝐵𝐶𝐷 is a square where points 𝐸 and 𝑂 lie on 𝐴𝐶, 𝐴𝐸=2cm, 𝐸𝑂=13cm, and 𝑂𝐶=1cm.

  • A 1 3 8
  • B 5 3 2 4
  • C 5 3 2 8
  • D 5 2 2 1

Q20:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵. Point 𝐷 lies on 𝐵𝐶 and point 𝐸 lies on 𝐴𝐶 where 𝐸𝐷∥𝐴𝐵. Find the area of the trapezoid 𝐴𝐵𝐷𝐸 given 𝐴𝐵=24cm, 𝐶𝐷=35cm, and 𝐸𝐷=12cm. Give the answer to two decimal places.

Q21:

Find the value of tantan𝐵𝐶 given 𝐴𝐵𝐶 is an isosceles triangle where 𝐴𝐵=𝐴𝐶=45cm and 𝐵𝐶=72cm.

  • A 9 1 6
  • B 6 4 9
  • C 1 5 1 6
  • D 1 6 9

Q22:

𝐴 𝐵 𝐶 is a triangle where 𝑏=11√2𝑎 and 𝑚∠𝐶=45∘. Find the value of tan𝐴 without using a calculator.

  • A 1 2 1 √ 2
  • B21
  • C 1 2 1
  • D √ 2 2 1

Q23:

In the given figure, the two triangles are similar.

Work out the value of tan𝜃 for △𝐴𝐵𝐶. Give your answer as a fraction in its simplest form.

  • A 1 1 1 2
  • B √ 2 6 5 1 1
  • C 1 2 1 1
  • D 1 2 √ 2 6 5 2 6 5
  • E 1 1 √ 2 6 5 2 6 5

Work out the value of tan𝜃 for △𝐷𝐸𝐹. Give your answer as a fraction in its simplest form.

  • A √ 2 6 5 1 2
  • B 1 2 1 1
  • C 1 2 √ 2 6 5 2 6 5
  • D 1 1 1 2
  • E 1 1 √ 2 6 5 2 6 5

What can be said about the value of tan𝜃 for two similar triangles?

  • AThey will never be equal.
  • BThey are always equal.
  • CThey are often equal.

Q24:

𝐴 𝐵 is a diametre of a circle with radius 17 cm. Point 𝐶 is on the circumference of the circle where 𝐴𝐶⊥𝐶𝐵 and 𝐴𝐶=16cm. Find the exact values of tan𝐴 and tan𝐵.

  • A t a n 𝐴 = 8 1 7 , t a n 𝐵 = 1 5 8
  • B t a n 𝐴 = 1 5 8 , t a n 𝐵 = 8 1 5
  • C t a n 𝐴 = 1 5 8 , t a n 𝐵 = 8 1 7
  • D t a n 𝐴 = 8 1 5 , t a n 𝐵 = 1 5 8

Q25:

Find the value of tantan𝑋+𝑍, given 𝑋𝑌𝑍 is a right-angled triangle at 𝑌 where 𝑋𝑌=12cm and 𝑋𝑍=15cm giving the answer to two decimal places.

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