Inverse trigonometric function equations

4

1. Solve the equation \(\sin ^ { - 1 } ( 3 x ) = - \frac { 1 } { 3 } \pi\), giving the solution in an exact form.
2. Solve, by factorising, the equation \(2 \cos \theta \sin \theta - 2 \cos \theta - \sin \theta + 1 = 0\) for \(0 \leqslant \theta \leqslant \pi\).

2 Find the value of \(x\) satisfying the equation \(\sin ^ { - 1 } ( x - 1 ) = \tan ^ { - 1 } ( 3 )\).

3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).

3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).

7. (a) For \(- \frac { \pi } { 2 } \leqslant y \leqslant \frac { \pi } { 2 }\), sketch the graph of \(y = \mathrm { g } ( x )\) where $$g ( x ) = \arcsin x \quad - 1 \leqslant x \leqslant 1$$ (b) Find the exact value of \(x\) for which $$3 g ( x + 1 ) + \pi = 0$$

2 Given that \(\arcsin x = \frac { 1 } { 6 } \pi\), find \(x\). Find \(\arccos x\) in terms of \(\pi\).

5 Given that \(\arcsin x = \frac { 1 } { 6 } \pi\), find \(x\). Find \(\arccos x\) in terms of \(\pi\).

1 Solve each of the following equations, giving your answers in exact form.

1. \(6 \arcsin x - \pi = 0\).
2. \(\arcsin x = \arccos x\).

3 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: let \(\arcsin x = \theta\).]

9 Given that \(\arcsin x = \frac { 1 } { 6 } \pi\), find \(x\). Find \(\arccos x\) in terms of \(\pi\).

10. (a) Given that \(- \frac { \pi } { 2 } < \mathrm { g } ( x ) < \frac { \pi } { 2 }\), sketch the graph of \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = \arctan x , \quad x \in \mathbb { R }$$ (b) Find the exact value of \(x\) for which $$3 g ( x + 1 ) - \pi = 0$$ The equation \(\arctan x - 4 + \frac { 1 } { 2 } x = 0\) has a positive root at \(x = \alpha\) radians.
(c) Show that \(5 < \alpha < 6\) The iteration formula $$x _ { n + 1 } = 8 - 2 \arctan x _ { n }$$ can be used to find an approximation for \(\alpha\) (d) Taking \(x _ { 0 } = 5\), use this formula to find \(x _ { 1 }\) and \(x _ { 2 }\), giving each answer to 3 decimal places.

7 Given that \(\arcsin x = \arccos y\), prove that \(x ^ { 2 } + y ^ { 2 } = 1\). [Hint: let \(\arcsin x = \theta\).] Section B (36 marks)

6 Solve each of the following equations, giving your answers in exact form.

1. \(6 \arcsin x - \pi = 0\).
2. \(\arcsin x = \arccos x\).