Mathematics Multiple Choice Questions on “Inverse Trigonometry”.
1. What will be the value of x + y + z if cos-1 x + cos-1 y + cos-1 z = 3π?
a) -1/3
b) 1
c) 3
d) -3
Answer: d
Clarification: The equation is cos-1 x + cos-1 y + cos-1 z = 3π
This means cos-1 x = π, cos-1 y = π and cos-1 z = π
This will be only possible when it is in maxima.
As, cos-1 x = π so, x = cos-1 π = -1 similarly, y = z = -1
Therefore, x + y + z = -1 -1 -1
So, x + y + z = -3.
2. Which value is similar to sin-1sin(6 π/7)?
a) sin-1(π/7)
b) cos-1(π/7)
c) sin-1(2π/7)
d) coses-1(π/7)
Answer: a
Clarification: sin-1sin(6 π/7)
Now, sin(6 π/7) = sin(π – 6 π/7)
= sin(2π + 6 π/7) = sin(π/7)
= sin(3π – 6 π/7) = sin(20π/7)
= sin(-π – 6 π/7) = sin(-15π/7)
= sin(-2π + 6 π/7) = sin(-8π/7)
= sin(-3π – 6 π/7) = sin(-27π/7)
Therefore, sin-1sin(6 π/7) = sin-1(π/7).
3. What is the value of sin-1(-x) for all x belongs to [-1, 1]?
a) -sin-1(x)
b) sin-1(x)
c) 2sin-1(x)
d) sin-1(-x)/2
Answer: a
Clarification: Let, θ = sin-1(-x)
So, -π/2 ≤ θ ≤ π/2
=> -x = sinθ
=> x = -sinθ
=> x = sin(-θ)
Also, -π/2 ≤ -θ ≤ π/2
=> -θ = sin-1(x)
=> θ = -sin-1(x)
So, sin-1(-x) = -sin-1(x)
4. What is the value of sin-1(sin 6)?
a) -2π – 6
b) 2π + 6
c) -2π + 6
d) 2π – 6
Answer: c
Clarification: We know, sin x = sin(π – x)
So, sin 6 = sin(π – 6)
= sin(2π + 6)
= sin(3π – 6)
= sin(-π – 6)
= sin(-2π – 6)
= sin(-3π – 6)
So, sin-1(sin 6) = sin-1(sin (-2π + 6))
= -2π + 6
5. What is the value of cos-1(-x) for all x belongs to [-1, 1]?
a) cos-1(-x)
b) π – cos-1(x)
c) π – cos-1(-x)
d) π + cos-1(x)
Answer: b
Clarification: Let, θ = cos-1(-x)
So, 0 ≤ θ ≤ π
=> -x = cosθ
=> x = -cosθ
=> x = cos(-θ)
Also, -π ≤ -θ ≤ 0
So, 0 ≤ π -θ ≤ π
=> -θ = cos-1(x)
=> θ = -cos-1(x)
So, cos-1(x) = π – θ
θ = π – cos-1(x)
=> cos-1(-x) = π – cos-1(x)
6. The given graph is for which equation?
a) y = sinx
b) y = sin-1x
c) y = cosecx
d) y = secx
Answer: b
Clarification: The following graph represents 2 equations.
The pink curve is the graph of y = sinx
The blue curve is the graph for y = sin-1x
This curve passes through the origin and approaches to infinity in both positive and negative axes.
7. The given graph is for which equation?
a) cosec-1x
b) secx
c) cos-1x
d) cotx
Answer: c
Clarification: There are 2 curves.
The green curve is the graph of y = cosx
The red curve is the graph for y = cos-1x
This curve origin from some point before π/3 and approaches to infinity in both positive y axis by intersecting at a point near 1.5 in y axis.