If sin theta = 1 /2 then find the value of theta .
The Correct Answer and Explanation is:
Given:sinθ=12\sin \theta = \frac{1}{2}sinθ=21
Correct Answer:θ=30∘orθ=150∘\theta = 30^\circ \quad \text{or} \quad \theta = 150^\circθ=30∘orθ=150∘
(or in radians: θ=π6orθ=5π6\theta = \frac{\pi}{6} \quad \text{or} \quad \theta = \frac{5\pi}{6}θ=6πorθ=65π)
Explanation
The sine function relates an angle in a right-angled triangle to the ratio of the length of the side opposite the angle to the hypotenuse. The value of sine is positive in both the first and second quadrants of the unit circle. To find the angle when sinθ=12\sin \theta = \frac{1}{2}sinθ=21, consider the standard angles used in trigonometry.
Looking at the unit circle, the sine of 30∘30^\circ30∘ is known to be exactly 12\frac{1}{2}21. This is derived from the properties of a 30-60-90 triangle, where the side ratios are 1 (opposite 30°), 3\sqrt{3}3 (opposite 60°), and 2 (the hypotenuse). Therefore:sin30∘=12\sin 30^\circ = \frac{1}{2}sin30∘=21
However, sine is also positive in the second quadrant. The angle in the second quadrant that has the same sine value is found using the identity:sin(180∘−θ)=sinθ\sin(180^\circ – \theta) = \sin \thetasin(180∘−θ)=sinθ
Thus:sin150∘=sin(180∘−30∘)=12\sin 150^\circ = \sin(180^\circ – 30^\circ) = \frac{1}{2}sin150∘=sin(180∘−30∘)=21
So, both 30∘30^\circ30∘ and 150∘150^\circ150∘ satisfy the equation. These are the principal values in the interval 0∘0^\circ0∘ to 360∘360^\circ360∘.
In radian measure, these angles convert to:30∘=π6,150∘=5π630^\circ = \frac{\pi}{6}, \quad 150^\circ = \frac{5\pi}{6}30∘=6π,150∘=65π
Because the sine function is periodic with a period of 360∘360^\circ360∘ (or 2π2\pi2π radians), other possible solutions can be expressed as:θ=30∘+360∘norθ=150∘+360∘n\theta = 30^\circ + 360^\circ n \quad \text{or} \quad \theta = 150^\circ + 360^\circ nθ=30∘+360∘norθ=150∘+360∘n
for any integer nnn, representing the full set of angles where sinθ=12\sin \theta = \frac{1}{2}sinθ=21.
