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Mathematics

What are double angle identities?

Double angle identities appear in both the AA SL and AA HL courses, but do not appear in either the AI SL or AI HL courses. There are double angle identities for sine and cosine, which appear in both the AA SL and AA HL courses, and a double angle identity for tangent, which appears only in the AA HL course. There is one double angle identity for the sine ratio - $\sin 2\theta = 2 \sin \theta \cos \theta$ There are three double angle identities for the cosine ratio - $\cos 2\theta = \cos ^2 \theta - \sin ^2 \theta$ - $\cos 2\theta = 2 \cos ^2 \theta - 1$ - $\cos 2\theta = 1 - 2 \sin ^2 \theta$ There is one double angle identity for the tangent ratio - $\tan 2\theta = \dfrac{2 \tan \theta}{1 - \tan ^2 \theta}$ **The double angle identities give us a way of expressing a trigonometric ratio in another form that may make it easier to solve an equation, or help us to find the exact value of a new angle, particularly in paper 1 where we cannot use a G.D.C. to help us.** For example, suppose we are asked to solve the equation $\sin \theta = \dfrac{1}{4 \cos \theta}$ for $0 \leq \theta \leq 2\pi$. We could rearrange this by multiplying both sides by $2 \cos \theta$, giving us $$ \begin{align*} \sin \theta \times 2 \cos \theta &= \dfrac{1}{4 \cos \theta} \times 2 \cos \theta \\[8pt] 2 \sin \theta \cos \theta &= \dfrac{1}{2} \end{align*} $$ Now on the left hand side we can apply the double angle identity for sine so that we have $$ \begin{align*} \sin 2\theta &= \dfrac{1}{2} \end{align*} $$ From here we can proceed as though solving a “normal” trigonometric equation, finding that the two solutions are $\theta = \dfrac{\pi}{12}$ and $\theta = \dfrac{5\pi}{12}$. As a second example, suppose we are asked to find the exact value of $\tan \left(\dfrac{\pi}{12}\right)$. From ratio triangles, we know that $\tan \left(\dfrac{\pi}{6}\right)=\dfrac{1}{\sqrt 3}$ and $\dfrac{\pi}{6}$ is double the angle $\dfrac{\pi}{12}$ so we could use the double angle identity for tangent to help us. We can set up our equation as follows: $$ \begin{align*} \tan 2\theta &= \frac{2 \tan \theta}{1 - \tan ^2 \theta} \\[12pt] \tan \left(\frac{\pi}{6}\right) &= \frac{2 \tan \frac{\pi}{12}}{1 - \tan ^2 \frac{\pi}{12}} \\[12pt] \frac{1}{\sqrt 3} &= \frac{2 \tan \frac{\pi}{12}}{1 - \tan ^2 \frac{\pi}{12}} \end{align*} $$ Now we could rearrange this to make a quadratic equation. $$ \begin{align*} \tan ^2 \left(\frac{\pi}{12}\right) + 2\sqrt{3} \tan \left(\frac{\pi}{12}\right) - 1 = 0 \end{align*} $$ We could then complete the square, or use the quadratic formula to find that $\tan\left(\dfrac{\pi}{12}\right) = \pm 2 - \sqrt {3}$. Then we can reason that $\dfrac{\pi}{12}$ is in quadrant 1 where the tangent function is positive, hence $\tan\left(\dfrac{\pi}{12}\right)$ must equal $2 - \sqrt{3}$. If faced with a trigonometric equation or expression in an exam that doesn’t appear to be solvable or easy to simplify in its current form,, it is useful to investigate whether a substitution using a double angle identity makes the equation more easily solvable, or the expression more easy to simplify.

Psychology

What is the right to withdraw in psychological studies?

The right to withdraw is one of a range of ethical considerations that researchers are required to follow when conducting psychological research. The right to withdraw specifically refers to the principle that a person should be able to remove themselves and their data from a research study at any time, without experiencing undue pressure or negative consequences. The right to withdraw, along with other ethical obligations, ensures that all research is conducted responsibly and that participants and the wider community are not adversely impacted as a result of the research. Participants should be informed of their right to withdraw when providing informed consent, and should be reminded of it again during debriefing.

Physics

What is the unit of force in the SI system?

The unit of force in the SI system is the newton. The symbol for the newton is N. The newton is what is referred to as a derived unit. Derived units are composed of a combination of the SI fundamental units. There are seven fundamental SI units: $\hspace{2em}$ metre (m) used for measuring length\ $\hspace{2em}$ second (s) used for measuring time\ $\hspace{2em}$ kilogram (kg) used for measuring mass\ $\hspace{2em}$ kelvin (K) used for measuring temperature\ $\hspace{2em}$ ampere (A) used for measuring electric current\ $\hspace{2em}$ mol (mol) used for measuring quantities\ $\hspace{2em}$ candela (cd) used for measuring luminous intensity Using our known physics equations, we can determine how to express the newton in fundamental SI units. Starting with Newton’s First Law, $\hspace{2em}\ F = ma$ We can substitute the units for the variables to get $\hspace{2em}$ 1 N = (1 kg)(1 m s$^{-2}$) Therefore the unit kg m s$^{-2}$ is equivalent to N

Environmental Systems and Societies

What is a primary consumer in an ecosystem?

Primary consumers are organisms that eat producers. They occupy the second trophic level in a food chain. Examples are **zooplankton**, which eat phytoplankton, **mountain gorillas**, which eat various parts of about 142 different plant species, and **grasshoppers**, which eat the leaves of grasses like wheat and rice. $\hspace{2em}$ **producer $\rightarrow$ primary consumer $\rightarrow$ secondary consumer $\rightarrow$ tertiary consumer** As you can see in the food chain above, producers are the first trophic level in a food chain, using photosynthesis to make carbon compounds that provide matter and energy for growth and survival. Consumers cannot photosynthesise, and so must obtain carbon compounds from other organisms. Primary consumers eat producers, and are eaten by secondary consumers. They, in turn, are eaten by tertiary consumers. Primary consumers play an important role in regulating the population sizes of both the primary producers they eat and the secondary consumers that eat them. Because consumers feed in such different ways, we have specific terms to describe them. The following terms are most relevant in IB ESS: - Herbivores are primary consumers that eat plants, e.g. koalas eat eucalyptus leaves. The trophic level of the consumers below is determined by the trophic level of the organisms they eat. - Predators eat prey; a predator such as an owl is a secondary consumer if its prey is a primary consumer, such as a mouse that eats corn, but a tertiary consumer if its prey is a secondary consumer, such as a frog that eats grasshoppers that eat grass. - Parasites such as mistletoe and tapeworms live in or on another organism and harm it as they feed on it, but do not usually kill it. - Scavengers like vultures eat dead organisms that were killed by other organisms or died of natural causes. - Decomposers such as soil bacteria, fungi and earthworms break down dead organisms or their parts as they feed on them. - Detritivores, such as earthworms, are decomposers that eat decomposing parts of organisms from all trophic levels, as well as their faeces. - Saprotrophs such as fungi are decomposers that break down decomposing parts of organisms by excreting enzymes onto them and then absorbing the products.

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