Question:
If Cos θ = 0.6, then Tan θ = ?
यदि Cos θ = 0.6, तो Tan θ = ?
Options:
- 0.8
- 4/3
- 1.25
- 3/4
- 0.8
- 4/3
- 1.25
- 3/4
Correct Answer: b)
Explanation:
Detailed Explanation:
We are given $\cos \theta = 0.6$ and need to find $\tan \theta$.
Step-by-step Solution:
- Convert $\cos \theta$ into a fraction:$$\cos \theta = 0.6 = \frac{6}{10} = \frac{3}{5}$$
- In a right-angled triangle:$$\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{3}{5}$$Let Base $= 3k$ and Hypotenuse $= 5k$.
- By the Pythagoras Theorem:$$\text{Perpendicular} = \sqrt{\text{Hypotenuse}^2 – \text{Base}^2} = \sqrt{(5k)^2 – (3k)^2} = \sqrt{25k^2 – 9k^2} = \sqrt{16k^2} = 4k$$
- Now, calculate $\tan \theta$:$$\tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{4k}{3k} = \frac{4}{3}$$
- Alternatively, using the identity $\sin \theta = \sqrt{1 – \cos^2 \theta}$:$$\sin \theta = \sqrt{1 – (0.6)^2} = \sqrt{1 – 0.36} = \sqrt{0.64} = 0.8$$$$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{0.8}{0.6} = \frac{8}{6} = \frac{4}{3}$$
Therefore, the value of $\tan \theta$ is 4/3. The correct option is b).
विस्तृत व्याख्या:
हमें $\cos \theta = 0.6$ दिया गया है और $\tan \theta$ का मान ज्ञात करना है।
चरण-दर-चरण हल:
- $\cos \theta$ को भिन्न के रूप में लिखने पर:$$\cos \theta = 0.6 = \frac{6}{10} = \frac{3}{5}$$
- समकोण त्रिभुज में:$$\cos \theta = \frac{\text{आधार}}{\text{कर्ण}} = \frac{3}{5}$$माना आधार $= 3$ और कर्ण $= 5$ है।
- पाइथागोरस प्रमेय के अनुसार लंब की गणना:$$\text{लंब} = \sqrt{\text{कर्ण}^2 – \text{आधार}^2} = \sqrt{5^2 – 3^2} = \sqrt{25 – 9} = \sqrt{16} = 4$$
- $\tan \theta$ का मान:$$\tan \theta = \frac{\text{लंब}}{\text{आधार}} = \frac{4}{3}$$
- वैकल्पिक रूप से: $\sin \theta = \sqrt{1 – \cos^2 \theta} = \sqrt{1 – 0.36} = \sqrt{0.64} = 0.8$।$$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{0.8}{0.6} = \frac{8}{6} = \frac{4}{3}$$
अतः $\tan \theta$ का मान 4/3 है। सही विकल्प b) है।