Fundamental relationships, simplification of expressions, reduction to the first quadrant and parity (chapters III and IV). Teste 41PUC-75 The value of the expression $25\operatorname{sen}^2 x - 9\operatorname{tg}^2 x$, knowing that $\operatorname{cossec} x = \frac{5}{4}$ and $x$ are from the first quadrant, is: a) $2$b) $3$c) $4$d) $0$e) $1$ $\operatorname{sen} x = \frac{4}{5}$, $\cos x = \frac{3}{5}$ e $\operatorname{tg} x = \frac{4}{3}$: $25 \cdot \frac{16}{25} - 9 \cdot \frac{16}{9} = 16 - 16 = 0$. Teste 42GV-76 If $\operatorname{sen} a = \frac{24}{25}$ and $\sec a$ is negative, then the value of $\sqrt{\dfrac{1 - \cos a}{1 + \cos a}}$ is: a) $\frac{3}{4}$b) $\frac{3}{5}$c) $\frac{5}{4}$d) $\frac{4}{3}$e) $\frac{1}{2}$ Negative secant means negative cosine: $\cos a = -\frac{7}{25}$. Then $\sqrt{\frac{1 + 7/25}{1 - 7/25}} = \sqrt{\frac{32}{18}} = \frac{4}{3}$.
The story The names even function and odd function come from the powers: $x^2$, $x^4$, $x^6$, … do not change when we change $x$ to $-x$, and $x$, $x^3$, $x^5$, … change signs. The terms appear in a work by the young Leonhard Euler on a curve problem proposed at the time, “reciprocal trajectories”, written around 1727. Euler also realized that the cosine behaves like even powers and the sine like odd powers, which is clear in the series he used: $\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots$ only has even powers, and $\operatorname{sen} x = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots$, only odd ones.
In calculation exercises, the fundamental relationship gives the module and the quadrant gives the sign. In the identities, indicate which of the three methods you used. Calculation of functions from one of them 73 Knowing that $\operatorname{sen} x = \dfrac{4}{5}$ and $\dfrac{\pi}{2} < x < \pi$, calculate the other circular functions of $x$. See the resolutionAs $\dfrac{\pi}{2} < x < \pi \Rightarrow \cos x < 0$, we have: $$\cos x = -\sqrt{1 - \operatorname{sen}^2 x} = -\sqrt{1 - \frac{16}{25}} = -\sqrt{\frac{9}{25}} = -\frac{3}{5}$$$$\operatorname{tg} x = \frac{\operatorname{sen} x}{\cos x} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3} \qquad \operatorname{cotg} x = \frac{\cos x}{\operatorname{sen} x} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4}$$$$\sec x = \frac{1}{\cos x} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} \qquad \operatorname{cossec} x = \frac{1}{\operatorname{sen} x} = \frac{1}{\frac{4}{5}} = \frac{5}{4}.$$ 74 Knowing that $\operatorname{cossec} x = -\dfrac{25}{24}$ and $\pi < x < \dfrac{3\pi}{2}$, calculate the other circular functions of $x$. Ver resposta$\operatorname{sen} x = -\dfrac{24}{25}$, $\cos x = -\dfrac{7}{25}$, $\operatorname{tg} x = \dfrac{24}{7}$, $\operatorname{cotg} x = \dfrac{7}{24}$, $\sec x = -\dfrac{25}{7}$
An identity is an equality that holds for all values where both sides make sense. Demonstrating identities is the great training in manipulating trigonometry: each step uses an already proven relationship. The story To demonstrate identities it is necessary to write the functions in short form. Until the 17th century this did not exist: each relationship was said in words — “the sine of the complement of the arc”, “the tangent of the bent arc” — and a one-line identity occupied a paragraph. The abbreviations appeared little by little: Gunter used sin in a figure in 1624, the Frenchman Pierre Hérigone printed it in a book in 1634, and the Englishman William Oughtred wrote something like s co arc for cosine. It was Euler, in Introductio (1748), who established the use of sin., cos., tang. as quantities that are added, multiplied and squared — and it is this use that allows you to manipulate an identity as an algebraic expression.