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 = 3 0 1 4 ′ 1 8 ′ ′ ∘ , 𝛽 = 5 4 2 0 ′ 4 1 ′ ′ ∘
  • B 𝛼 = 5 4 2 0 ′ 4 1 ′ ′ ∘ , 𝛽 = 3 5 3 9 ′ 1 9 ′ ′ ∘
  • C 𝛼 = 5 4 2 0 ′ 4 1 ′ ′ ∘ , 𝛽 = 3 0 1 4 ′ 1 8 ′ ′ ∘
  • 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 𝐴 𝐵 = 4 5 c m and 𝐵 𝐶 = 6 8 c m . Find the measure of ∠ 𝐶 giving the answer to the nearest second.

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

Q3:

𝐴 𝐵 𝐶 𝐷 is a rhombus where 𝐴 𝐶 = 2 4 c m and 𝐵 𝐷 = 3 2 c m . Find 𝑚 ∠ 𝐵 𝐴 𝐶 giving the answer to the nearest second.

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

Q4:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝐴 𝐵 = 1 9 c m and 𝑚 ∠ 𝐶 = 5 4 ∘ . Find the length of 𝐵 𝐶 giving the answer to two decimal places.

Q5:

𝐴 𝐵 𝐶 is a triangle where 𝐴 𝐷 ⟂ 𝐵 𝐶 , 𝑚 ∠ 𝐵 = 5 1 1 6 ′ 1 8 ′ ′ ∘ , 𝐵 𝐶 = 2 0 c m , and 𝐴 𝐷 = 8 c m . Find 𝑚 ∠ 𝐶 giving the answer to the nearest second.

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

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 𝐴 𝐵 and the 𝑥 -axis giving the answer to the nearest second.

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

Q7:

A flagpole 5.9 metres 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 4 3 7 ∘
  • C 6 1 4 0 ∘
  • D 2 5 2 3 ∘

Q8:

Find the value of 𝐾 , given 𝐴 𝐵 𝐶 is an equilateral triangle, where point 𝐷 lies on 𝐴 𝐵 , 𝐴 𝐷 = 5 c m , 𝐷 𝐵 = 1 2 c m , and 𝐾 𝑋 = √ 3 t a n .

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

Q9:

Find 𝑥 to two decimal places.

Q10:

In the given figure, 𝑚 ∠ 𝐵 𝐴 𝐶 = 9 0 ∘ and 𝐴 𝐷 ⊥ 𝐵 𝐶 . What is 𝐴 𝐶 𝜃 t a n ?

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

Q11:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 , where 𝐴 𝐵 = 6 4 c m . Point 𝐷 lies on 𝐵 𝐶 and point 𝐸 lies on 𝐴 𝐶 where 𝐸 𝐷 ∥ 𝐴 𝐵 . Find the length of 𝐷 𝐵 given 𝐶 𝐷 = 1 2 c m and 𝐸 𝐷 = 3 5 c m . Give the answer to two decimal places.

Q12:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝑚 ∠ 𝐶 = 3 4 1 2 ′ ∘ and 𝐵 𝐶 = 2 5 c m . Find the length of 𝐴 𝐵 giving the answer to two decimal places.

Q13:

Find the exact value of t a n 𝐶 .

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

Q14:

The dimensions of a rectangular garden is 13 metres by 12 metres. Triangle 𝐴 𝐸 𝐶 represents the lawn where 𝐸 𝐶 = 1 0 m . Find the measure of ∠ 𝐸 𝐴 𝐵 giving the answer to the nearest second.

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

Q15:

𝑋 𝑌 𝑍 is a triangle where 𝐿 lies on 𝑋 𝑍 such that 𝑌 𝐿 ⟂ 𝑋 𝑍 , t a n t a n 𝑍 = 𝑋 𝑌 𝐿 = 2 0 2 1 and 𝑍 𝐿 = 2 6 c m . Find the area of 𝑋 𝑌 𝑍 giving the answer to two decimal places.

Q16:

Find the length of 𝐷 𝐶 given 𝐴 𝐵 𝐶 𝐷 is a quadrilateral where 𝐴 𝐷 = 6 c m and 𝐴 𝐵 = 8 c m .

  • A 4 4 5 cm
  • B 7 1 2 cm
  • C 10 cm
  • D 1 3 1 3 cm

Q17:

𝐴 𝐵 𝐶 is a triangle where 𝐴 𝐷 ⟂ 𝐵 𝐶 , 𝐴 𝐷 = 1 6 c m , 𝑚 ∠ 𝐵 = 4 4 ∘ , and 𝑚 ∠ 𝐶 = 4 8 ∘ . Find, to the nearest centimetre, the length of 𝐵 𝐶 .

Q18:

In the given figure, 𝑚 ∠ 𝐵 𝐴 𝐶 = 9 0 ∘ where 𝐴 𝐷 ⊥ 𝐵 𝐶 . What is 𝐴 𝐷 𝜃 t a n ?

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

Q19:

Find the value of t a n t a n 𝑥 + 𝑦 given 𝐴 𝐵 𝐶 𝐷 is a square where points 𝐸 and 𝑂 lie on 𝐴 𝐶 , 𝐴 𝐸 = 2 c m , 𝐸 𝑂 = 1 3 c m , and 𝑂 𝐶 = 1 c m .

  • A 5 3 2 4
  • B 1 3 8
  • 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 𝐵 𝐶 and point 𝐸 lies on 𝐴 𝐶 where 𝐸 𝐷 ∥ 𝐴 𝐵 . Find the area of the trapezium 𝐴 𝐵 𝐷 𝐸 given 𝐴 𝐵 = 2 4 c m , 𝐶 𝐷 = 3 5 c m , and 𝐸 𝐷 = 1 2 c m . Give the answer to two decimal places.

Q21:

𝐴 𝐵 𝐶 is a right angled triangle at 𝐵 . Find the area of 𝐴 𝐵 𝐶 given that 𝐴 𝐵 = 2 4 c m , 𝐷 ∈ 𝐴 𝐶 , 𝐸 ∈ 𝐵 𝐶 , 𝐷 𝐸 ⟂ 𝐵 𝐶 , and 5 𝐷 𝐸 = 6 𝐸 𝐶 . Round your answer to two decimal places.

Q22:

Find the value of t a n t a n 𝐵 𝐶 given 𝐴 𝐵 𝐶 is an isosceles triangle where 𝐴 𝐵 = 𝐴 𝐶 = 4 5 c m and 𝐵 𝐶 = 7 2 c m .

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

Q23:

𝐴 𝐵 𝐶 is a triangle where 𝑏 = 1 1 √ 2 𝑎 and 𝑚 ∠ 𝐶 = 4 5 ∘ . Find the value of t a n 𝐴 without using a calculator.

  • A √ 2 2 1
  • B 1 2 1 √ 2
  • C 21
  • D 1 2 1

Q24:

In the given figure, the two triangles are similar.

Work out the value of t a n 𝜃 for △ 𝐴 𝐵 𝐶 . Give your answer as a fraction in its simplest form.

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

Work out the value of t a n 𝜃 for △ 𝐷 𝐸 𝐹 . Give your answer as a fraction in its simplest form.

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

What can be said about the value of t a n 𝜃 for two similar triangles?

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

Q25:

𝐴 𝐵 is a diametre of a circle with radius 17 cm. Point 𝐶 is on the circumference of the circle where 𝐴 𝐶 ⊥ 𝐶 𝐵 and 𝐴 𝐶 = 1 6 c m . Find the exact values of t a n 𝐴 and t a n 𝐵 .

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

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