General Solutions of Simple Trigonometric Equations
Principal value of an Angle (or) principal solution of an equation
Definition I:
There exists only one value \( \alpha \) \( \alpha \) for \( \theta \) in \(\left[ {\frac{{ - \pi }} {2},\frac{\pi } {2}} \right]\) satisfying \( \sin \theta = K = \sin \alpha \) \( ( - 1 \leqslant k \leqslant 1) \) or \( ( - 1 \leqslant \sin \alpha \leqslant 1) \)
Then the value \( \alpha \) is called Principal value of \( \theta \) (or) the principal solution of \(sin \theta = K = \sin \alpha ,\,\,\,\,\alpha \in \left[ {\frac{{ - \pi }} {2},\frac{\pi } {2}} \right] \) .
Note : \( \exists \) a unique value \( \alpha \) for \( \theta \) in \(\alpha\in\left[ {\frac{{ - \pi }} {2},\frac{\pi } {2}} \right]\) satisfying \( \cos ec\theta = K = \cos ec\alpha \) is called principal value of \( \theta (\theta \ne 0) \).
Definition 2 :- \( \exists \) only one value of \( \alpha \) for \( \theta \) in \( \alpha \in \left[ {0,\pi } \right] \) satisfying sec\(\theta\)=\( \cos \theta = K = \cos \alpha ( - 1 \leqslant k \leqslant 1) \), the value ‘\( \alpha \)’ is called the principal value of \( \theta \) (or) the principal solution of \( \cos \theta = K = \cos \alpha \).
Note : \( \exists \) a unique value \( \alpha \) for \( \theta \) in \( \alpha \in \left[ {0,\pi } \right] \) where \( \theta \ne \frac{\pi } {2} \) satisfying \( \sec \theta = K(|k| \geqslant 1;\,\,\,\, - 1 - \geqslant k \geqslant 1) \), then the value of \( \alpha \) is called principal value of \( \theta \) (or) principal solution of \( \sec \theta = K \)
Definition 3 :- \( \exists \) only one value of \( \alpha \) for \( \theta \) in \(\alpha\in\left( {\frac{{ - \pi }} {2},\frac{\pi } {2}} \right)\) satisfying \( \tan \theta = k = \tan \alpha (K \in R) \), Then the value ‘\( \alpha \)’ is called the principal value of (or) the principal solution of \( \tan \theta = k \).
Note : \( \exists \) a unique value \( \alpha \) for \( \theta \) in \(\alpha\in\left( {\frac{{ - \pi }} {2},\frac{\pi } {2}} \right)\) where \( \theta \ne 0 \) satisfying
\( \cot \theta = k = \cot \alpha ,\,\,\,\,\,\,(K \in R) \) , then the value of is called principal value of (or) principal solution of \( \cot \theta = k \).