Correct Answer - Option 3 :
\(\frac{2\pi}{3}\)
Concept:
Values of Trigonometric Ratios for Common Angles:
| |
0°
|
30°
|
45°
|
60°
|
90°
|
|
sin
|
0
|
\(\frac{1}{2}\)
|
\(\frac{1}{\sqrt2}\)
|
\(\frac{\sqrt3}{2}\)
|
1
|
|
cos
|
1
|
\(\frac{\sqrt3}{2}\)
|
\(\frac{1}{\sqrt2}\)
|
\(\frac{1}{2}\)
|
0
|
|
tan
|
0
|
\(\frac{1}{\sqrt3}\)
|
1
|
√3
|
∞
|
|
csc
|
∞
|
2
|
√2
|
\(\frac{2}{\sqrt3}\)
|
1
|
|
sec
|
1
|
\(\frac{1}{\sqrt3}\)
|
√2
|
2
|
∞
|
|
cot
|
∞
|
√3
|
1
|
\(\frac{1}{\sqrt3}\)
|
0
|
Inverse Trigonometric Functions for Negative Arguments:
|
sin-1 (-x)
|
- sin-1 x
|
cos-1 (-x)
|
π - cos-1 x
|
|
csc-1 (-x)
|
- csc-1 x
|
sec-1 (-x)
|
π - sec-1 x
|
|
tan-1 (-x)
|
- tan-1 x
|
cot-1 (-x)
|
π - cot-1 x
|
Calculation:
From the properties of inverse trigonometric functions, we get:
\(\rm \cos^{-1}\left(-\frac12\right)\) = \(\rm \pi-\cos^{-1}\left(\frac12\right)\)
Using the table of trigonometric values, we see that \(\rm \cos^{-1}\left(\frac12\right)\) = 60∘.
∴ \(\rm \cos^{-1}\left(-\frac12\right)\) = π - 60∘. = \(\rm \pi-\frac\pi3=\frac{2\pi}{3}\)
