ARCHIES HIGHER SECONDARY SCHOOL
INVERSE TRIGONOMETRIC FUNCTION
SUBJECT: MATHEMATICS CLASS: XII
1. Evaluate the following :
(i) 𝑠𝑖𝑛 (𝑠𝑖𝑛10) Ans: 𝟑𝝅 − 𝟏𝟎
(ii) 𝑡𝑎𝑛 (tan (−4)) Ans: 𝝅 − 𝟒
(iii) 𝑐𝑜𝑠 (𝑐𝑜𝑠12) − 𝑠𝑖𝑛 (𝑠𝑖𝑛12) Ans: 𝟖𝝅 − 𝟐𝟒
(iv) 𝑠𝑖𝑛 (𝑠𝑖𝑛2) Ans: 𝝅 − 𝟐
(v) 𝑠𝑖𝑛 (𝑠𝑖𝑛5) Ans: 𝟓 − 𝟐𝝅
(vi) 𝑡𝑎𝑛 (𝑡𝑎𝑛4) Ans: 𝟒 − 𝝅
(vii) 𝑐𝑜𝑠 (𝑐𝑜𝑠10) Ans: 𝟒𝝅 − 𝟏𝟎.
(viii) 𝑡𝑎𝑛 𝑡𝑎𝑛(−6) Ans: 𝟐𝝅 − 𝟔
2. Prove that: 𝑐𝑜𝑠 = 2𝑡𝑎𝑛 𝑡𝑎𝑛 . 𝑡𝑎𝑛
.
3. Prove that: 𝑡𝑎𝑛 + 𝑐𝑜𝑠 + 𝑡𝑎𝑛 − 𝑐𝑜𝑠 = .
4.If 𝑦 = 𝑐𝑜𝑡 √𝑐𝑜𝑠𝑥 − 𝑡𝑎𝑛 √𝑐𝑜𝑠𝑥 , prove that 𝑠𝑖𝑛𝑦 = 𝑡𝑎𝑛 .
5. Prove that 𝑐𝑜𝑠 𝑡𝑎𝑛 𝑠𝑖𝑛(𝑐𝑜𝑡 𝑥) =
6. If 𝑠𝑖𝑛 𝑥 + 𝑡𝑎𝑛 𝑥= , prove that 2𝑥 + 1 = √5.
7. Prove that : 𝑠𝑒𝑐 (𝑡𝑎𝑛 2) + 𝑐𝑜𝑠𝑒𝑐 (𝑐𝑜𝑡 3) = 15.
8. If 𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦 + 𝑡𝑎𝑛 𝑧 = 𝜋 , then prove that 𝑥 + 𝑦 + 𝑧 = 𝑥𝑦𝑧.
9. Solve the following equations for 𝑥:
(i) 𝑠𝑖𝑛 (1 − 𝑥 ) − 2𝑠𝑖𝑛 𝑥 = Ans: 𝒙 = 𝟎
𝟏
(ii) 𝑠𝑖𝑛 (6𝑥) + 𝑠𝑖𝑛 6√3𝑥 = − Ans: 𝒙 = − 𝟏𝟐
𝟏
(iii) 𝑡𝑎𝑛 2𝑥 + 𝑡𝑎𝑛 3𝑥 = Ans: 𝒙 =
𝟔
𝟏
(iv) 𝑡𝑎𝑛 (𝑥 + 1) + 𝑡𝑎𝑛 (𝑥 − 1) = 𝑡𝑎𝑛 Ans: 𝒙 =
𝟒
𝟏 𝟏
(v) 𝑡𝑎𝑛 (𝑥 − 1) + 𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 (𝑥 + 1) = 𝑡𝑎𝑛 3𝑥 Ans: 𝒙 = 𝟎, 𝟐 , − 𝟐
10. If (𝑡𝑎𝑛 𝑥) + (𝑐𝑜𝑡 𝑥) = , then find 𝑥. Ans: 𝒙 = −𝟏.
𝝅𝟐
11. Find the minimum value of (𝑡𝑎𝑛 𝑥) + (𝑐𝑜𝑡 𝑥 ) . Ans:
𝟖
12. If 𝑐𝑜𝑠 𝑥 + 𝑐𝑜𝑠 𝑦 + 𝑐𝑜𝑠 𝑧 = 𝜋 , prove that: 𝑥 + 𝑦 + 𝑧 + 2𝑥𝑦𝑧 = 1.
13. If 𝑐𝑜𝑠 + 𝑐𝑜𝑠 = 𝛼 , prove that 9𝑥 − 12𝑥𝑦𝑐𝑜𝑠𝛼 + 4𝑦 = 36𝑠𝑖𝑛 𝛼.
14. If 𝑠𝑖𝑛 𝑥 + 𝑠𝑖𝑛 𝑦 + 𝑠𝑖𝑛 𝑧 = , then find the value of 𝑥 + 𝑦 + 𝑧 + 2𝑥𝑦𝑧 Ans: 1
15. If 𝑐𝑜𝑠 𝑥 + 𝑐𝑜𝑠 𝑦 + 𝑐𝑜𝑠 𝑧 = 3𝜋 , then find the values of
(i) 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥 Ans: 3
(ii) (𝑥 + 𝑦)(𝑦 + 𝑧)(𝑧 + 𝑥)Ans: −𝟖
(iii) 𝑥 +𝑦 +𝑧 − Ans: 6.
16. If 𝑠𝑖𝑛 𝑥 + 𝑠𝑖𝑛 𝑦 + 𝑠𝑖𝑛 𝑧 = , then find the values of
(i) 𝑥 + 𝑦 + 𝑧 Ans: 3
(ii) 𝑥 +𝑦 +𝑧 − 3𝑥𝑦𝑧 Ans:0
17. Find the domain of the function
(i) 𝑐𝑜𝑠 √1 − 𝑥 Ans:[0,1]
(ii) 𝑠𝑖𝑛 √𝑥 − 1 Ans:[1,2]
(iii) 𝑐𝑜𝑠 (−𝑥 ) Ans:[−𝟏, 𝟏]
(iv) 𝑠𝑖𝑛 (𝑥 − 4) Ans: −√𝟓, −√𝟑 ∪ √𝟑, √𝟓
𝟏 𝟏
(v) 𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠 (2𝑥) Ans: − ,
𝟐 𝟐
(vi) 𝑠𝑖𝑛 (−𝑥 ) Ans:[−𝟏, 𝟏]
(vii) 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠𝑥 Ans: [−𝟏, 𝟏]
(viii) 𝑠𝑖𝑛 √𝑥 − 1 Ans: −√𝟐, −𝟏 ∪ 𝟏, √𝟐
(ix) 𝑐𝑜𝑠 (|𝑥 − 1|) Ans: [0,2]
𝟏 𝟏
(x) 2𝑐𝑜𝑠 2𝑥 + 𝑠𝑖𝑛 𝑥 Ans: − 𝟐 , 𝟐
𝟐
(xi) 𝑠𝑒𝑐 (3𝑥 − 1) Ans: ]−∞, 𝟎] ∪ ,∞
𝟑
𝝅
18. (i) Draw the graph of 𝑐𝑜𝑠 𝑥, 𝑥 ∈ [−1,0]. Also , write its range. Ans: ,𝝅
𝟐
𝝅 𝝅
(ii) Draw the graph of 𝑠𝑖𝑛 𝑥, 𝑥 ∈ − , . Also , write its range. Ans: − ,
√ √ 𝟒 𝟒
19. Write in simplest form :
𝝅 𝟏
(i) 𝑐𝑜𝑡 √1 + 𝑥 − 𝑥 Ans: 𝟒 + 𝟐 𝒕𝒂𝒏 𝟏𝒙
𝟏 𝟑
(ii) 𝑐𝑜𝑠 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 Ans: 𝒄𝒐𝒔 − 𝒙.
𝟓
20. If 𝑡𝑎𝑛 + 𝑡𝑎𝑛 +……..+𝑡𝑎𝑛 = 𝑡𝑎𝑛 𝑥 , then find the value of 𝑥.
. . ( )
𝒏
Ans 𝒙 = .
𝒏 𝟐
21. If 𝛼 ≤ 𝑡𝑎𝑛 𝑥 + 𝑐𝑜𝑡 𝑥 + 𝑠𝑖𝑛 𝑥 ≤ 𝛽 then prove that : 𝛼 = 0, 𝛽 = 𝜋.
22. Which is greater , 𝑡𝑎𝑛1 or 𝑡𝑎𝑛 1 ? Ans: 𝒕𝒂𝒏𝟏 > 𝒕𝒂𝒏 𝟏 𝟏
23. Find the minimum value of n for which 𝑡𝑎𝑛 > , 𝑛 ∈ 𝑁. Ans: 𝒏 = 𝟒
24. If 𝑐𝑜𝑠 𝑥 > 𝑠𝑖𝑛 𝑥, then
(a) < 𝑥 ≤ 1 (b) 0 ≤ 𝑥 <
√ √
(c) −1 ≤ 𝑥 < (d) 𝑥 > 0
√
Ans: C
