Step 1: Simplify the given equation
3 cot θ + 4 = 0
3 cot θ = -4
cot θ = -4/3
Step 2: Find tan θ
Since tan θ = 1/cot θ,
tan θ = -3/4
Step 3: Determine values of sin θ and cos θ
Since θ is in the fourth quadrant, cos θ is positive and sin θ is negative.
Using the reference triangle with opposite = 3 and adjacent = 4:
Hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5
Therefore:
sin θ = -3/5
cos θ = 4/5
Step 4: Calculate sin 2θ and cos 2θ using double-angle formulas
sin 2θ = 2 sin θ cos θ
sin 2θ = 2 × (-3/5) × (4/5) = -24/25
cos 2θ = cos² θ - sin² θ
cos 2θ = (4/5)² - (-3/5)² = 16/25 - 9/25 = 7/25
Step 5: Final Calculation
sin 2θ + cos 2θ = (-24/25) + (7/25)
sin 2θ + cos 2θ = -17/25
Final Answer: -17/25 (or -0.68)
Similar Questions
- If $x = \sec\theta - \tan\theta$ and $y = \text{cosec}\theta + \cot\theta$, then which one of the following is correct ?
Important Questions from Trigonometric Ratios and Identities
The value of 4 sin 230° + 3 cot 260° - 2 tan 245° is:
The value of 1 - sin 35° cos 55° is equal to:
If sin 3 θ = cos ( θ – 6°), then θ is:
If θ = 45°, then what will be the value of \(\frac{{\\sin \,\theta \, + \,\cos \,\theta }}{{\sin \,\theta \, - \,\cos \,\theta }}\) ?
If sin A = \(\frac{1}{2}\) and cos B = \(\frac{1}{2}\) then find A + B.
