Published by:
CGP EDU Academic Team
Published on: September 12, 2026
Find the values of the following T-ratios
(i)
(ii)
(iii) 
Text Solution
Verified by ExpertsThe correct answer is:
A
To find the T-ratios, we will evaluate each trigonometric function step-by-step.
1. **Finding \( \csc 315^{\circ} \)**:
The cosecant function is defined as \( \csc \theta = \frac{1}{\sin \theta} \).
Since \( 315^{\circ} = 360^{\circ} - 45^{\circ} \), we know that \( \sin 315^{\circ} = -\sin 45^{\circ} = -\frac{\sqrt{2}}{2} \).
Therefore, \( \csc 315^{\circ} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}. \)
2. **Finding \( \cos 210^{\circ} \)**:
Using the unit circle, \( 210^{\circ} = 180^{\circ} + 30^{\circ} \). Thus, \( \cos 210^{\circ} = -\cos 30^{\circ} = -\frac{\sqrt{3}}{2}. \)
3. **Finding \( \sin(-330^{\circ}) \)**:
The sine function is odd, which means \( \sin(-\theta) = -\sin(\theta) \). Therefore, \( \sin(-330^{\circ}) = -\sin(330^{\circ}) \).
Since \( 330^{\circ} = 360^{\circ} - 30^{\circ} \), we find that \( \sin 330^{\circ} = -\sin 30^{\circ} = -\frac{1}{2}. \) Hence, \( \sin(-330^{\circ}) = -(-\frac{1}{2}) = \frac{1}{2}. \)
Combining all results:
\( \csc 315^{\circ} = -\sqrt{2}, \cos 210^{\circ} = -\frac{\sqrt{3}}{2}, \sin(-330^{\circ}) = \frac{1}{2}. \)
These T-ratios provide the required values.
1. **Finding \( \csc 315^{\circ} \)**:
The cosecant function is defined as \( \csc \theta = \frac{1}{\sin \theta} \).
Since \( 315^{\circ} = 360^{\circ} - 45^{\circ} \), we know that \( \sin 315^{\circ} = -\sin 45^{\circ} = -\frac{\sqrt{2}}{2} \).
Therefore, \( \csc 315^{\circ} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}. \)
2. **Finding \( \cos 210^{\circ} \)**:
Using the unit circle, \( 210^{\circ} = 180^{\circ} + 30^{\circ} \). Thus, \( \cos 210^{\circ} = -\cos 30^{\circ} = -\frac{\sqrt{3}}{2}. \)
3. **Finding \( \sin(-330^{\circ}) \)**:
The sine function is odd, which means \( \sin(-\theta) = -\sin(\theta) \). Therefore, \( \sin(-330^{\circ}) = -\sin(330^{\circ}) \).
Since \( 330^{\circ} = 360^{\circ} - 30^{\circ} \), we find that \( \sin 330^{\circ} = -\sin 30^{\circ} = -\frac{1}{2}. \) Hence, \( \sin(-330^{\circ}) = -(-\frac{1}{2}) = \frac{1}{2}. \)
Combining all results:
\( \csc 315^{\circ} = -\sqrt{2}, \cos 210^{\circ} = -\frac{\sqrt{3}}{2}, \sin(-330^{\circ}) = \frac{1}{2}. \)
These T-ratios provide the required values.
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