Question

If $4{{\sin }^{2}}\theta -1=0$ and angle is less than $90{}^\circ $, find the value of θ.

Answer 1

Hint: The value of trigonometric ratios is always calculated in a right-angled triangle.
There are $6$ main trigonometric ratios in all.

Complete step by step answer:
The trigonometry is a Greek word whose meaning is derived as:
“Tri” means three while “Gon” means sides and “Metron” means measure. It helps in studying the relation between sides and angles of the triangle.
The main values at which trigonometric ratios are given and easy to calculate:
 $0{}^\circ $,$30{}^\circ $,$45{}^\circ $,$60{}^\circ $and $90{}^\circ $. The value of trigonometric ratios at multiples of these numbers is also easy to determine.
Trigonometric ratio in terms of sides of triangle are given as:
sinθ = P/H, cosθ = B/H, tanθ = P/B, cosecθ = H/P, secθ = H/B, cotθ = B/P
→ H = Hypotenuse, P = Perpendicular, B = Base
In a right-angled triangle, the Pythagoras Theorem is applied to get the value of ratios.
Pythagoras Theorem is indicated by equation:
H2 = P2 + B2
As per the given equation,
\[\begin{align}
  & 4{{\sin }^{2}}\theta -1=0 \\
 & {{\sin }^{2}}\theta =\dfrac{1}{4} \\
 & \sin \theta =\sqrt{\dfrac{1}{4}} \\
 & \sin \theta =\pm \dfrac{1}{2}
\end{align}\]
As θ is less than $90{}^\circ $, the angle θ lies in the first quadrant which means that sinθ is positive and is equal to $\dfrac{1}{2}$.
$\begin{align}
  & \sin \theta =\dfrac{1}{2} \\
 & \theta =30{}^\circ
\end{align}$
This indicates that the value of θ is equal to $30{}^\circ $.

Note: There are three pairs of trigonometric ratios – (sin, cosec), (cos, sec) and (tan, cot).
Cosec and sin are inversely related to each other. Similarly, pair-wise cos and sec and tan and cot are inversely related to each other.
The trigonometric ratio is written as sinθ which means the ratio of perpendicular to Hypotenuse but it does not mean product of sin and θ.