Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.
Start Practicing

Worksheet: Inverse Trigonometric Functions

Q1:

The given tables show some values of 𝑓 ( π‘₯ ) , 𝑔 ( π‘₯ ) , and β„Ž ( π‘₯ ) . Which function corresponds to the function t a n βˆ’ 1 π‘₯ ?

π‘₯ βˆ’ √ 3 βˆ’ 1 0 1 √ 3 1
𝑓 ( π‘₯ ) βˆ’ πœ‹ 3 βˆ’ πœ‹ 4 0 πœ‹ 6 πœ‹ 4
π‘₯ βˆ’ √ 3 βˆ’ 1 2 0 √ 3 1
𝑔 ( π‘₯ ) βˆ’ πœ‹ 6 βˆ’ πœ‹ 4 0 πœ‹ 6 πœ‹ 2
π‘₯ βˆ’ 1 βˆ’ 1 2 0 1 √ 3 √ 3
β„Ž ( π‘₯ ) βˆ’ πœ‹ 4 βˆ’ πœ‹ 6 0 πœ‹ 6 πœ‹ 3
  • A β„Ž ( π‘₯ )
  • B 𝑔 ( π‘₯ )
  • C 𝑓 ( π‘₯ )

Q2:

Find the exact value of t a n βˆ’ 1 ( 1 ) , in radians in terms of πœ‹ , which is located in the 1st quadrant.

  • A πœ‹ 2
  • B1
  • C0
  • D πœ‹ 4
  • E πœ‹ 3

Q3:

Find the exact value of t a n βˆ’ 1 ( 0 ) in radians.

Q4:

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

Table A

π‘₯ 0 πœ‹ 6 πœ‹ 4 πœ‹ 3 πœ‹ 2 2 πœ‹ 3 3 πœ‹ 4 5 πœ‹ 6 πœ‹
s i n π‘₯ 0 1 2 √ 2 2 √ 3 2 1 √ 3 2 √ 2 2 1 2 0

Table B

π‘₯ βˆ’ πœ‹ 2 βˆ’ πœ‹ 3 βˆ’ πœ‹ 4 βˆ’ πœ‹ 6 0 πœ‹ 6 πœ‹ 4 πœ‹ 3 πœ‹ 2
s i n π‘₯ βˆ’ 1 βˆ’ √ 3 2 βˆ’ √ 2 2 βˆ’ 1 2 0 1 2 √ 2 2 √ 3 2 1
  • A Table B
  • B Table A

Q5:

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

Table A

π‘₯ 0 πœ‹ 6 πœ‹ 4 πœ‹ 3 πœ‹ 2 2 πœ‹ 3 3 πœ‹ 4 5 πœ‹ 6 πœ‹
c o s π‘₯ 1 √ 3 2 √ 2 2 1 2 0 βˆ’ 1 2 βˆ’ √ 2 2 βˆ’ √ 3 2 βˆ’ 1

Table B

π‘₯ βˆ’ πœ‹ 2 βˆ’ πœ‹ 3 βˆ’ πœ‹ 4 βˆ’ πœ‹ 6 0 πœ‹ 6 πœ‹ 4 πœ‹ 3 πœ‹ 2
c o s π‘₯ 0 1 2 √ 2 2 √ 3 2 1 √ 3 2 √ 2 2 1 2 0
  • A Table A
  • B Table B

Q6:

Find the exact value of c o s s i n ο€Ό ο€Ό 5 1 3   βˆ’ 1 .

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

Q7:

Find the exact value of s i n s i n βˆ’ 1 ο€Ό ο€Ό 4 πœ‹ 3   in radians.

  • A πœ‹ 3
  • B βˆ’ 5 πœ‹ 3
  • C βˆ’ 4 πœ‹ 3
  • D βˆ’ πœ‹ 3
  • E πœ‹

Q8:

Find the exact value of c o s c o s βˆ’ 1 ο€» πœ‹ 7  in radians.

  • A 8 πœ‹ 7
  • B 6 πœ‹ 7
  • C βˆ’ 8 πœ‹ 7
  • D πœ‹ 7
  • E πœ‹

Q9:

Find the exact value of c o t c o t βˆ’ 1 ο€Ό 4 πœ‹ 3  in radians.

  • A πœ‹
  • B βˆ’ πœ‹ 3
  • C βˆ’ 4 πœ‹ 3
  • D πœ‹ 3

Q10:

Find the exact value of c o s s i n ο€Ό ο€Ό βˆ’ 4 5   βˆ’ 1 .

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

Q11:

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 ∘

Q12:

Find the exact value of s i n s i n βˆ’ 1 ο€Ό ο€Ό βˆ’ 5 πœ‹ 6   in radians.

  • A 5 πœ‹ 6
  • B πœ‹ 6
  • C πœ‹
  • D βˆ’ πœ‹ 6

Q13:

Find the exact value of t a n c o t βˆ’ 1 βˆ’ 1 ο€Ό 3 5  + ο€Ό 3 5  in radians.

  • A πœ‹ 4
  • B πœ‹ 3
  • C πœ‹ 6
  • D πœ‹ 2
  • E πœ‹

Q14:

Find the exact value of s i n c o s βˆ’ 1 βˆ’ 1 ο€Ό 5 1 3  + ο€Ό 5 1 3  in radians.

  • A πœ‹ 4
  • B πœ‹ 3
  • C πœ‹ 6
  • D πœ‹ 2
  • E πœ‹

Q15:

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

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

Q16:

Find the exact value of c o s c o s βˆ’ 1 ο€» βˆ’ πœ‹ 1 0  in radians.

  • A βˆ’ 9 πœ‹ 1 0
  • B 9 πœ‹ 1 0
  • C 1 0 πœ‹ 1 1
  • D πœ‹ 1 0
  • E πœ‹

Q17:

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

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

Q18:

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

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

Q19:

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.

Q20:

𝐴 𝐡 𝐢 is an isosceles triangle where 𝐴 𝐡 = 𝐴 𝐢 and 𝐡 𝐢 = 2 4 c m . The point 𝐷 lies on 𝐡 𝐢 where 𝐴 𝐷 βŸ‚ 𝐡 𝐢 and 𝐴 𝐷 = 1 6 c m . Find the length of 𝐴 𝐢 giving the answer to the nearest centimeter and the value of ∠ 𝐢 giving the answer to the nearest second.

  • A 𝐴 𝐢 = 1 2 c m , π‘š ∠ 𝐢 = 5 3 7 β€² 4 8 β€² β€² ∘
  • B 𝐴 𝐢 = 2 0 c m , π‘š ∠ 𝐢 = 3 6 5 2 β€² 1 2 β€² β€² ∘
  • C 𝐴 𝐢 = 1 6 c m , π‘š ∠ 𝐢 = 3 6 5 2 β€² 1 2 β€² β€² ∘
  • D 𝐴 𝐢 = 2 0 c m , π‘š ∠ 𝐢 = 5 3 7 β€² 4 8 β€² β€² ∘