Trigonometric Graphs & Equations Flashcards

All 17 cards in this deck

What is the period of $y = \sin x$ and $y = \cos x$?

$360°$ — each graph repeats every $360°$.

What is the period of $y = \tan x$?

$180°$ — the graph repeats every $180°$.

For $0° \le x \le 360°$, where are the asymptotes of $y = \tan x$?

$x = 90°$ and $x = 270°$, where $\cos x = 0$ so $\tan x$ is undefined.

How do you find the value of a trigonometric ratio of an angle outside $0°$ to $360°$?
e.g. $\sin 400°$

1. Add or subtract $360°$ until the angle lies in $0°$ to $360°$: $400° - 360° = 40°$
2. The ratio is unchanged: $\sin 400° = \sin 40°$

How many solutions does $\sin x = k$ have in $0° \le x \le 360°$ when $0 < k < 1$?

Two — the line $y = k$ cuts the sine curve twice, at $\alpha$ and $180° - \alpha$.

State the identity linking $\tan\theta$ with $\sin\theta$ and $\cos\theta$.

$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

State the identity connecting $\sin^2\theta$ and $\cos^2\theta$.

$\sin^2\theta + \cos^2\theta = 1$

True or false? $\sin^2\theta$ means $\sin(\theta^2)$.

False. $\sin^2\theta$ means $(\sin\theta)^2$ — square the value of $\sin\theta$.

How do you find $\cos\theta$ exactly when you are given $\sin\theta$ and $\theta$ is acute?
e.g. $\sin\theta = \frac{3}{5}$

1. Use $\sin^2\theta + \cos^2\theta = 1$: $\cos^2\theta = 1 - \left(\frac{3}{5}\right)^2 = \frac{16}{25}$
2. Square root, choosing the sign for the quadrant: $\cos\theta = \frac{4}{5}$

What are the steps to prove a trigonometric identity?
e.g. prove $\tan\theta\cos\theta \equiv \sin\theta$

1. Start with the more complicated side: LHS $= \tan\theta\cos\theta$
2. Replace $\tan\theta$ by $\frac{\sin\theta}{\cos\theta}$: $\frac{\sin\theta}{\cos\theta}\times\cos\theta$
3. Simplify to reach the other side: $=\sin\theta=$ RHS

How do you solve an equation of the form $a\sin x = b\cos x$?
e.g. $\sin x = \cos x$ for $0° \le x \le 360°$

1. Divide both sides by $\cos x$ to get a single ratio: $\tan x = 1$
2. Solve, adding $180°$ for the second solution: $x = 45°$ and $x = 225°$

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: $x = 0°,\ 180°,\ 360°,\ 210°,\ 330°$

True or false? $\cos(-\theta) = -\cos\theta$ for any angle $\theta$.

False. $\cos(-\theta) = \cos\theta$, because the graph of $y = \cos x$ is symmetrical about the $y$-axis.