Worksheet: Inverse Trigonometric Functions

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In this worksheet, we will practice calculating exact values of trigonometric inverses and recognizing the inverse of trigonometric functions from tables.

Q1:

The given tables show some values of 𝑓 ( 𝑥 ) , 𝑔 ( 𝑥 ) , and ℎ ( 𝑥 ) . Which function corresponds to the function s i n − 1 𝑥 ?

𝑥 − √ 3 2 − 1 2 0 1 2 √ 2 2
𝑓 ( 𝑥 ) − 𝜋 3 − 𝜋 6 0 𝜋 4 𝜋 3
𝑥 − √ 3 2 − √ 2 2 0 √ 3 2 1
𝑔 ( 𝑥 ) − 𝜋 6 − 𝜋 4 0 𝜋 6 𝜋 2
𝑥 − 1 − 1 2 0 √ 2 2 √ 3 2
ℎ ( 𝑥 ) − 𝜋 2 − 𝜋 6 0 𝜋 4 𝜋 3
  • A 𝑔 ( 𝑥 )
  • B 𝑓 ( 𝑥 )
  • C ℎ ( 𝑥 )

Q2:

The given tables show some values of 𝑓 ( 𝑥 ) , 𝑔 ( 𝑥 ) , and ℎ ( 𝑥 ) . Which function corresponds to the function t a n − 1 𝑥 ?

𝑥 − √ 3 − 1 0 1 √ 3 1
𝑓 ( 𝑥 ) − 𝜋 3 − 𝜋 4 0 𝜋 6 𝜋 4
𝑥 − √ 3 − 1 2 0 √ 3 1
𝑔 ( 𝑥 ) − 𝜋 6 − 𝜋 4 0 𝜋 6 𝜋 2
𝑥 − 1 − 1 2 0 1 √ 3 √ 3
ℎ ( 𝑥 ) − 𝜋 4 − 𝜋 6 0 𝜋 6 𝜋 3
  • A ℎ ( 𝑥 )
  • B 𝑔 ( 𝑥 )
  • C 𝑓 ( 𝑥 )

Q3:

Find the exact value of c o s s i n 5 1 3 − 1 .

  • A 1 3 1 2
  • B 5 1 3
  • C 1 3 5
  • D 1 2 1 3
  • E 5 1 2

Q4:

Find the exact value of t a n ( 1 ) , in radians in terms of 𝜋 , which is located in the 1st quadrant.

  • A 𝜋 2
  • B1
  • C0
  • D 𝜋 4
  • E 𝜋 3

Q5:

Find the exact value of t a n − 1 ( 0 ) in radians.

Q6:

Which of these two value tables shows a domain of the sine function that can be used to construct its inverse function?

Table A

𝑥 0 𝜋 6 𝜋 4 𝜋 3 𝜋 2 2 𝜋 3 3 𝜋 4 5 𝜋 6 𝜋
s i n 𝑥 0 1 2 √ 2 2 √ 3 2 1 √ 3 2 √ 2 2 1 2 0

Table B

𝑥 − 𝜋 2 − 𝜋 3 − 𝜋 4 − 𝜋 6 0 𝜋 6 𝜋 4 𝜋 3 𝜋 2
s i n 𝑥 − 1 − √ 3 2 − √ 2 2 − 1 2 0 1 2 √ 2 2 √ 3 2 1
  • A Table B
  • B Table A

Q7:

Which of these two value tables shows a domain of the cosine function that can be used to construct its inverse function?

Table A

𝑥 0 𝜋 6 𝜋 4 𝜋 3 𝜋 2 2 𝜋 3 3 𝜋 4 5 𝜋 6 𝜋
c o s 𝑥 1 √ 3 2 √ 2 2 1 2 0 − 1 2 − √ 2 2 − √ 3 2 − 1

Table B

𝑥 − 𝜋 2 − 𝜋 3 − 𝜋 4 − 𝜋 6 0 𝜋 6 𝜋 4 𝜋 3 𝜋 2
c o s 𝑥 0 1 2 √ 2 2 √ 3 2 1 √ 3 2 √ 2 2 1 2 0
  • A Table A
  • B Table B

Q8:

Find the exact value of s i n s i n − 1 4 𝜋 3 in radians.

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

Q9:

Find the exact value of c o s c o s − 1 𝜋 7 in radians.

  • A 8 𝜋 7
  • B 6 𝜋 7
  • C − 8 𝜋 7
  • D 𝜋 7
  • E 𝜋

Q10:

Find the exact value of c o t c o t − 1 4 𝜋 3 in radians.

  • A 𝜋
  • B − 𝜋 3
  • C − 4 𝜋 3
  • D 𝜋 3

Q11:

Find the exact value of c o s s i n − 4 5 − 1 .

  • A 5 2
  • B 4 5
  • C 5 4
  • D 3 5
  • E 3 4

Q12:

Which of the following is closest to a r c t a n ( 1 0 0 0 0 ) ?

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

Q13:

Find the exact value of s i n s i n − 1 − 5 𝜋 6 in radians.

  • A 5 𝜋 6
  • B 𝜋 6
  • C 𝜋
  • D − 𝜋 6

Q14:

Find the exact value of t a n c o t − 1 − 1 3 5 + 3 5 in radians.

  • A 𝜋 4
  • B 𝜋 3
  • C 𝜋 6
  • D 𝜋 2
  • E 𝜋

Q15:

Find the exact value of s i n c o s − 1 − 1 5 1 3 + 5 1 3 in radians.

  • A 𝜋 4
  • B 𝜋 3
  • C 𝜋 6
  • D 𝜋 2
  • E 𝜋

Q16:

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

  • A 𝛼 = 3 5 5 0 ′ 1 6 ′ ′ ∘ , 𝛽 = 4 6 1 4 ′ 1 8 ′ ′ ∘
  • B 𝛼 = 4 6 1 4 ′ 1 8 ′ ′ ∘ , 𝛽 = 4 3 4 5 ′ 4 2 ′ ′ ∘
  • C 𝛼 = 4 6 1 4 ′ 1 8 ′ ′ ∘ , 𝛽 = 3 5 5 0 ′ 1 6 ′ ′ ∘
  • D 𝛼 = 4 3 4 5 ′ 4 2 ′ ′ ∘ , 𝛽 = 4 6 1 4 ′ 1 8 ′ ′ ∘

Q17:

Find the exact value of c o s c o s − 1 − 𝜋 1 0 in radians.

  • A − 9 𝜋 1 0
  • B 9 𝜋 1 0
  • C 1 0 𝜋 1 1
  • D 𝜋 1 0
  • E 𝜋

Q18:

𝐴 𝐵 𝐶 is a triangle where c o s 𝐴 = 0 . 2 3 2 7 and t a n 𝐵 = 0 . 9 3 8 1 . Find the value of ∠ 𝐶 giving the answer to the nearest minute.

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

Q19:

𝐴 𝐵 𝐶 𝐷 is a parallelogram with an area of 546 cm2. The point 𝐸 is on 𝐵 𝐶 where 𝐴 𝐸 ⟂ 𝐵 𝐶 , the ratio between 𝐵 𝐸 and 𝐸 𝐶 is 1 3 : and 𝐴 𝐸 = 4 2 c m . Find the measure of ∠ 𝐶 giving the answer to the nearest second.

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

Q20:

A man stands 5.4 m from a vertical wall. A footlight on the ground, 2.7 m from where he is standing, is switched on. If the man is 1.8 m tall and his shadow is 5.5 m tall, what angle does the light make with the horizontal? Give your answer to two decimal places.

Q21:

Find the exact value of t a n t a n − 1 4 𝜋 3 , in radians in terms of 𝜋 , which is located in the 1st quadrant.

  • A − 4 𝜋 3
  • B 2 𝜋 3
  • C − 𝜋 3
  • D 𝜋 3
  • E 𝜋

Q22:

Find the exact value of s i n s i n − 1 𝜋 3 in radians.

  • A 4 𝜋 3
  • B 𝜋 6
  • C 𝜋
  • D 𝜋 3
  • E − 𝜋 3

Q23:

Find the exact value of s i n − 1 ( 1 ) in radians.

  • A 3 𝜋 2
  • B − 𝜋 4
  • C − 𝜋 2
  • D 𝜋 2
  • E 𝜋 4

Q24:

Find the exact value of s i n s i n − 1 − 1 3 5 + 4 5 in radians.

  • A 𝜋 4
  • B 𝜋 3
  • C 𝜋 6
  • D 𝜋 2
  • E 𝜋

Q25:

Find the exact value of c s c c s c − 1 − 𝜋 9 in radians.

  • A 8 𝜋 9
  • B 𝜋 9
  • C − 1 0 𝜋 9
  • D − 𝜋 9
  • E − 1 7 𝜋 9

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