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
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Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
