To find the value of \(\cot\theta\) given \(\cos\theta = \frac{24}{25}\), we will use trigonometric identities and relationships. The formulae we will use are:
- \(\sin^2\theta + \cos^2\theta = 1\)
- \(\cot\theta = \frac{\cos\theta}{\sin\theta}\)
Step 1: Calculate \(\sin\theta\) using the Pythagorean identity:
Since we know that:
\(\cos\theta = \frac{24}{25}\)
We can substitute into \(\sin^2\theta + \cos^2\theta = 1\):
\(\sin^2\theta + \left(\frac{24}{25}\right)^2 = 1\)
\(\sin^2\theta + \frac{576}{625} = 1\)
\(\sin^2\theta = 1 - \frac{576}{625}\)
\(\sin^2\theta = \frac{625 - 576}{625}\)
\(\sin^2\theta = \frac{49}{625}\)
Taking the square root, since we assume \(\theta\) is in the first quadrant where \(\sin\theta\) is positive:
\(\sin\theta = \frac{7}{25}\)
Step 2: Calculate \(\cot\theta\) using the relation \(\cot\theta = \frac{\cos\theta}{\sin\theta}\):
\(\cot\theta = \frac{\frac{24}{25}}{\frac{7}{25}}\)
\(\cot\theta = \frac{24}{25} \times \frac{25}{7}\)
\(\cot\theta = \frac{24}{7}\)
Therefore, the value of \(\cot\theta\) is \(\frac{24}{7}\).
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Important Questions from Trigonometric Ratios
If \(cosec~\theta =\frac{29}{21}\) where 0 < θ < 90°, then what is the value of 4 sec θ + 4 tan θ?
The value of \(\cot \left( {cose{c^{ - 1}}\frac{5}{3} + {{\tan }^{ - 1}}\frac{2}{3}\;} \right)\)
The distance of the highest point on the graph of the function y = √3 cos x + sin x from the x-axis is:
If A = π / 6 and B = π / 3, then consider the following statements:
I. sin A + sin B = cos A + cos B
II. tan A + tan B = cot A + cot B
Which of the above statements is / are correct?
If \(\sin A = \frac{1}{{\sqrt 2 }}\) and \({\mathop{\rm Cos}\nolimits} B = \frac{{\sqrt 3 }}{2}\), then, find the value of (A + B)º.
A. 60º
B. 75º
C. 105º
D. 90º
