NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 8: Introduction to Trignometry and Its Applications - Exercise 8.3Question 12
Class 10 - Mathematics - CHAPTER 8: Introduction to Trignometry and Its Applications - Exercise 8.3
Question. 12
If \(2\sin^2\theta-\cos^2\theta=2\), find \(\theta\).
Answer:
\(\theta=90^\circ\).
Handwritten Notes
Video Explanation:
Detailed Answer with Explanation:
Step 1: Recall the basic trigonometric identity:
\(\sin^2\theta + \cos^2\theta = 1\).
This means \(\cos^2\theta = 1 - \sin^2\theta\).
Step 2: Substitute \(\cos^2\theta = 1 - \sin^2\theta\) into the given equation:
\(2\sin^2\theta - (1 - \sin^2\theta) = 2\).
Step 3: Simplify the equation:
\(2\sin^2\theta - 1 + \sin^2\theta = 2\).
\(3\sin^2\theta - 1 = 2\).
Step 4: Add 1 to both sides:
\(3\sin^2\theta = 3\).
Step 5: Divide both sides by 3:
\(\sin^2\theta = 1\).
Step 6: Take square root on both sides:
\(\sin\theta = \pm 1\).
Step 7: Now check where sine is equal to 1 or -1:
- \(\sin\theta = 1\) when \(\theta = 90^\circ\).
- \(\sin\theta = -1\) when \(\theta = 270^\circ\).
Step 8: Since we are usually asked for the acute principal angle, the required solution is:
\(\theta = 90^\circ\).