Solving Trig Equations Flashcards

All 5 cards in this deck

For angles from $0°$ to $360°$, which trigonometric ratios are positive in each quadrant?

$0°$–$90°$: all positive; $90°$–$180°$: $\sin$ only; $180°$–$270°$: $\tan$ only; $270°$–$360°$: $\cos$ only.

If $\cos x = k$ and $x = \alpha$ is one solution, what is the other solution in $0° \le x \le 360°$?
e.g. $\alpha = 20°$

$360° - \alpha$, so $x = 340°$.

True or false? If $x = \alpha$ is a solution of $\tan x = k$, then $x = \alpha + 180°$ is also a solution.

True. $\tan x$ has period $180°$, so solutions are $180°$ apart.

What are the steps to solve $\sin x = k$ in $0° \le x \le 360°$?
e.g. $\sin x = 0.5$

1. First solution: $\alpha = \sin^{-1}(0.5) = 30°$
2. Second solution: $180° - \alpha = 150°$
3. Check the interval (sketch the graph): $x = 30°$ and $x = 150°$

What are the steps to solve a quadratic trigonometric equation by factorising?
e.g. $2\sin^2 x + \sin x = 0$ for $0° \le x \le 360°$

1. Factorise, treating $\sin x$ as the unknown: $\sin x(2\sin x + 1) = 0$
2. Set each factor to zero: $\sin x = 0$ or $\sin x = -0.5$
3. Solve each in the interval: $x = 0°,\ 180°,\ 360°,\ 210°,\ 330°$