Worksheet: Trigonometric Ratios in Right Triangles

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In this worksheet, we will practice finding and express the values of the three trigonometric ratios—sine, cosine, and tangent—for a given angle in a right triangle.

Q1:

Find the main trigonometric ratios of ∠ 𝐵 given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐶 where 𝐴 𝐵 = 3 0 c m and 𝐵 𝐶 = 1 8 c m .

  • A s i n 𝐵 = 3 5 , c o s 𝐵 = 4 5 , t a n 𝐵 = 4 3
  • B s i n 𝐵 = 4 5 , c o s 𝐵 = 3 5 , t a n 𝐵 = 3 4
  • C s i n 𝐵 = 4 5 , c o s 𝐵 = 3 5 , t a n 𝐵 = 4 3
  • D s i n 𝐵 = 3 5 , c o s 𝐵 = 4 5 , t a n 𝐵 = 3 4

Q2:

𝐴 𝐵 is a diameter of a circle with radius 62.5 cm. Point 𝐶 is on the circumference of the circle where 𝐴 𝐶 ⟂ 𝐶 𝐵 and 𝐴 𝐶 = 7 5 c m . Find the exact values of c o s 𝐴 and s i n 𝐵 .

  • A c o s 𝐴 = 4 5 and s i n 𝐵 = 3 5
  • B c o s 𝐴 = 3 5 and s i n 𝐵 = 4 5
  • C c o s 𝐴 = 3 5 and s i n 𝐵 = 3 5
  • D c o s 𝐴 = 4 5 and s i n 𝐵 = 4 5

Q3:

Find the main trigonometric ratios of ∠ 𝐶 given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 2 0 𝐴 = 2 1 t a n .

  • A s i n 𝐶 = 2 1 2 9 , c o s 𝐶 = 2 0 2 9 and t a n 𝐶 = 2 0 2 1
  • B s i n 𝐶 = 2 1 2 9 , c o s 𝐶 = 2 0 2 9 and t a n 𝐶 = 2 1 2 0
  • C s i n 𝐶 = 2 0 2 9 , c o s 𝐶 = 2 1 2 9 and t a n 𝐶 = 2 1 2 0
  • D s i n 𝐶 = 2 0 2 9 , c o s 𝐶 = 2 1 2 9 and t a n 𝐶 = 2 0 2 1

Q4:

Find the value of c o s s i n 𝑥 5 + 𝑥 5 .

Q5:

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

Q6:

Find 𝑚 ∠ 𝐵 given 𝐴 𝐵 𝐶 is a triangle where 𝑚 ∠ 𝐴 = 1 1 1 ∘ and s i n c o s 𝐶 = 𝐶 .

Q7:

Find 𝑚 ∠ 𝐴 given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where √ 2 𝐴 𝐵 = 𝐴 𝐶 .

Q8:

Find c o t 𝛼 given 𝐴 𝐵 𝐶 𝐷 is a rectangle where t a n 𝜃 = 1 0 1 7 and 𝐵 𝐹 ⟂ 𝐴 𝐸 .

  • A 1 7 1 0
  • B 1 0 1 7
  • C − 1 7 1 0
  • D − 1 0 1 7

Q9:

Find the value of 3 1 𝜃 + 2 6 𝜃 s i n c o s .

  • A 2 6 + 5 𝜃 s i n
  • B 2 6 + 5 𝜃 c o s
  • C 5 7 + 𝜃 c o s
  • D 5 7 + 𝜃 s i n

Q10:

Find the value of c s c ( 2 7 0 − 𝜃 ) ∘ given s i n ( 9 0 − 𝜃 ) = 2 7 ∘ where 𝜃 is the smallest positive angle.

  • A 7 3 √ 5
  • B 7 2
  • C − 7 3 √ 5
  • D − 7 2

Q11:

Find the value of 7 1 6 5 + 7 1 6 5 c o s s i n ∘ ∘ .

Q12:

Simplify ( 𝜃 + 𝜃 ) − 2 𝜃 𝜃 s i n c o s s i n c o s .

Q13:

Simplify s i n c o s s i n c o s 𝜃 − 𝜃 𝜃 − 𝜃 .

Q14:

Find the value of ( 𝑥 + 𝑥 ) + ( 𝑥 − 𝑥 ) c o s s i n c o s s i n .

Q15:

Find c o t 𝐵 given that 𝐴 𝐵 𝐶 is a right triangle at 𝐶 , where 𝐴 𝐶 = 1 2 c m and 𝐵 𝐶 = 9 c m .

  • A 3 4
  • B 4 3
  • C 5 3
  • D 4 5
  • E 3 5

Q16:

Find the main trigonometric ratios of ∠ 𝐴 given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where the ratio between 𝐴 𝐵 and 𝐴 𝐶 is 4 ∶ 5 .

  • A s i n 𝐴 = 3 5 , c o s 𝐴 = 3 4 , t a n 𝐴 = 4 5
  • B s i n 𝐴 = 3 4 , c o s 𝐴 = 4 5 , t a n 𝐴 = 3 5
  • C s i n 𝐴 = 4 5 , c o s 𝐴 = 3 5 , t a n 𝐴 = 3 4
  • D s i n 𝐴 = 3 5 , c o s 𝐴 = 4 5 , t a n 𝐴 = 3 4

Q17:

Find s i n c o s 𝐶 𝐶 given that 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 𝐴 𝐵 = 8 c m and 𝐴 𝐶 = 1 7 c m .

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

Q18:

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

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

Q19:

In the triangle shown, is the side labelled 𝑦 adjacent to angle 𝜃 , opposite angle 𝜃 , or the hypotenuse?

  • Ahypotenuse
  • Badjacent
  • Copposite

Q20:

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

Q21:

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

  • A 𝑥 = 2 6 . 1 1 0 c m , 𝑦 = 3 8 . 2 8 5 c m
  • B 𝑥 = 4 7 . 4 3 2 c m , 𝑦 = 3 8 . 2 8 5 c m
  • C 𝑥 = 3 8 . 2 8 5 c m , 𝑦 = 2 6 . 1 1 0 c m
  • D 𝑥 = 3 8 . 2 8 5 c m , 𝑦 = 4 7 . 4 3 2 c m

Q22:

𝑋 𝑌 𝑍 is a right-angled triangle at 𝑌 , where 𝑋 𝑌 = 1 6 . 5 c m , 𝑌 𝑍 = 2 8 c m , and 𝑋 𝑍 = 3 2 . 5 c m . Find the measure of ∠ 𝑍 giving the answer to the nearest second.

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

Q23:

Find the value of s i n c o s 𝐵 + 𝐶 , given that 𝐴 𝐵 𝐶 is a triangle, where 𝐴 𝐵 = 𝐴 𝐶 = 4 1 c m , 𝐵 𝐶 = 1 8 c m , and 𝐴 𝐷 is drawn perpendicular to 𝐵 𝐶 intersecting at 𝐷 .

  • A 1 , 1 8 0 3 6 9
  • B 3 6 9 9 8 2
  • C 4 9 4 1
  • D 4 1 4 9

Q24:

Find the value of 6 2 5 𝐴 𝐴 s i n c o s given 𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 where 7 𝐴 − 2 4 = 0 t a n .

  • A175
  • B625
  • C168
  • D600

Q25:

𝐸 is a point inside a square 𝐴 𝐵 𝐶 𝐷 with a side length of 48 cm, where 𝐵 𝐸 = 𝐶 𝐸 and 𝑂 𝐸 = 3 0 c m . Find the value of 𝐾 , given that 𝐾 ( 𝑋 − 𝑋 ) = 1 3 0 c o s s i n .

  • A6
  • B42
  • C 1 4 2
  • D 1 6

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