If sin θ = \(\frac{12}{13}\) then what is the value of (tan θ + sec θ)2 (cosec θ - cot θ)-2 0 < θ < \(\frac{\pi }{2}\)
Trigonometry Problem: Evaluating an Expression with sin θ
We are given that \(\sin \theta = \frac{12}{13}\) and that \(0 < \theta < \frac{\pi}{2}\). We need to find the value of the expression \((\tan \theta + \sec \theta)^2 (\text{cosec } \theta - \cot \theta)^{-2}\).
Since \(0 < \theta < \frac{\pi}{2}\), \(\theta\) is in the first quadrant. In the first quadrant, all trigonometric ratios are positive.
Finding Other Trigonometric Ratios
We know \(\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}\). Let's consider a right-angled triangle where the angle is \(\theta\).
- Opposite side = 12
- Hypotenuse = 13
Using the Pythagorean theorem, the Adjacent side squared + Opposite side squared = Hypotenuse squared.
Adjacent side\(^2 + 12^2 = 13^2\)
Adjacent side\(^2 + 144 = 169\)
Adjacent side\(^2 = 169 - 144\)
Adjacent side\(^2 = 25\)
Adjacent side = \(\sqrt{25} = 5\) (since length must be positive).
Now we can find the other trigonometric ratios for \(\theta\):
- \(\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{5}{13}\)
- \(\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{12}{5}\)
- \(\text{cosec } \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{12}{13}} = \frac{13}{12}\)
- \(\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{5}{13}} = \frac{13}{5}\)
- \(\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{12}{5}} = \frac{5}{12}\)
Evaluating the Expression Terms
The expression is \((\tan \theta + \sec \theta)^2 (\text{cosec } \theta - \cot \theta)^{-2}\). Let's evaluate each part:
First Term: \((\tan \theta + \sec \theta)^2\)
Substitute the values of \(\tan \theta\) and \(\sec \theta\):
\(\tan \theta + \sec \theta = \frac{12}{5} + \frac{13}{5} = \frac{12 + 13}{5} = \frac{25}{5} = 5\)
So, \((\tan \theta + \sec \theta)^2 = (5)^2 = 25\).
Second Term: \((\text{cosec } \theta - \cot \theta)^{-2}\)
Substitute the values of \(\text{cosec } \theta\) and \(\cot \theta\):
\(\text{cosec } \theta - \cot \theta = \frac{13}{12} - \frac{5}{12} = \frac{13 - 5}{12} = \frac{8}{12}\)
Simplify the fraction \(\frac{8}{12}\) by dividing both numerator and denominator by 4:
\(\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}\)
Now, we need to calculate \((\frac{2}{3})^{-2}\). Remember that \(a^{-n} = \frac{1}{a^n}\) or \((\frac{a}{b})^{-n} = (\frac{b}{a})^n\).
\((\frac{2}{3})^{-2} = (\frac{3}{2})^2 = \frac{3^2}{2^2} = \frac{9}{4}\).
Calculating the Final Value
Now multiply the values of the two terms we calculated:
Value of expression = \((\tan \theta + \sec \theta)^2 \times (\text{cosec } \theta - \cot \theta)^{-2}\)
Value of expression = \(25 \times \frac{9}{4}\)
Value of expression = \(\frac{25 \times 9}{4} = \frac{225}{4}\).
Thus, the value of the expression is \(\frac{225}{4}\).
Revision Table: Key Trigonometric Identities
| Identity | Description |
|---|---|
| \(\sin^2 \theta + \cos^2 \theta = 1\) | Pythagorean Identity relating sine and cosine. |
| \(1 + \tan^2 \theta = \sec^2 \theta\) | Pythagorean Identity relating tangent and secant. |
| \(1 + \cot^2 \theta = \text{cosec}^2 \theta\) | Pythagorean Identity relating cotangent and cosecant. |
| \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) | Ratio identity for tangent. |
| \(\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta}\) | Ratio and reciprocal identity for cotangent. |
| \(\sec \theta = \frac{1}{\cos \theta}\) | Reciprocal identity for secant. |
| \(\text{cosec } \theta = \frac{1}{\sin \theta}\) | Reciprocal identity for cosecant. |
Additional Information: Understanding Quadrants
The quadrant in which the angle \(\theta\) lies is important because it determines the sign of the trigonometric ratios. The problem states \(0 < \theta < \frac{\pi}{2}\), which means \(\theta\) is in the first quadrant.
- Quadrant I (\(0 < \theta < \frac{\pi}{2}\)): All trigonometric ratios (sin, cos, tan, cosec, sec, cot) are positive.
- Quadrant II (\(\frac{\pi}{2} < \theta < \pi\)): Sine and cosecant are positive; others are negative.
- Quadrant III (\(\pi < \theta < \frac{3\pi}{2}\)): Tangent and cotangent are positive; others are negative.
- Quadrant IV (\(\frac{3\pi}{2} < \theta < 2\pi\)): Cosine and secant are positive; others are negative.
In this specific problem, knowing that \(\theta\) is in the first quadrant confirmed that \(\cos \theta\) and all other derived values would be positive, consistent with our calculations based on the right triangle sides.
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