Reduction to the 1st quadrant

117

Reduce to the 1st quadrant:
a) $\cos 178^\circ$ b) $\operatorname{sen} 251^\circ$ c) $\operatorname{tg} 290^\circ$
d) $\operatorname{cotg}\dfrac{7\pi}{6}$ e) $\sec 1924^\circ$ f) $\operatorname{cossec}\dfrac{23\pi}{6}$
g) $\operatorname{sen}\left(-\dfrac{7\pi}{6}\right)$ h) $\cos\left(-\dfrac{5\pi}{3}\right)$ i) $\operatorname{tg}\left(-\dfrac{3\pi}{4}\right)$
j) $\operatorname{sen}\dfrac{21\pi}{4}$ k) $\cos\dfrac{31\pi}{6}$ l) $\operatorname{tg}\dfrac{11\pi}{3}$
Ver resposta

a) $-\cos 2^\circ$ · b) $-\operatorname{sen} 71^\circ$ · c) $-\operatorname{tg} 70^\circ$ · d) $\operatorname{cotg}\dfrac{\pi}{6}$ · e) $-\sec 56^\circ$ · f) $-\operatorname{cossec}\dfrac{\pi}{6}$ · g) $\operatorname{sen}\dfrac{\pi}{6}$ · h) $\cos\dfrac{\pi}{3}$ · i) $\operatorname{tg}\dfrac{\pi}{4}$ · j) $-\operatorname{sen}\dfrac{\pi}{4}$ · k) $-\cos\dfrac{\pi}{6}$ · l) $-\operatorname{tg}\dfrac{\pi}{3}$

118

Reduce to the range $\left[0, \dfrac{\pi}{4}\right]$:
a) $\operatorname{sen} 261^\circ$ b) $\cos 2861^\circ$ c) $\operatorname{tg} 511^\circ$
d) $\operatorname{sen}\dfrac{4\pi}{3}$ e) $\cos\dfrac{2\pi}{3}$ f) $\operatorname{tg}\dfrac{5\pi}{3}$
g) $\operatorname{sen}\dfrac{5\pi}{6}$ h) $\cos\dfrac{7\pi}{6}$ i) $\operatorname{tg}\dfrac{11\pi}{6}$
j) $\operatorname{sen}\left(-\dfrac{5\pi}{3}\right)$ k) $\cos\left(-\dfrac{4\pi}{3}\right)$ l) $\operatorname{tg}\left(-\dfrac{11\pi}{3}\right)$
Ver resposta

a) $-\cos 9^\circ$ · b) $\cos 19^\circ$ · c) $-\operatorname{tg} 29^\circ$ · d) $-\cos\dfrac{\pi}{6}$ · e) $-\operatorname{sen}\dfrac{\pi}{6}$ · f) $-\operatorname{cotg}\dfrac{\pi}{6}$ · g) $\operatorname{sen}\dfrac{\pi}{6}$ · h) $-\cos\dfrac{\pi}{6}$ · i) $-\operatorname{tg}\dfrac{\pi}{6}$ · j) $\cos\dfrac{\pi}{6}$ · k) $-\operatorname{sen}\dfrac{\pi}{6}$ · l) $\operatorname{cotg}\dfrac{\pi}{6}$

Reduction Identities

119

Simplify the expressions: a) $\operatorname{sen}\left(\dfrac{\pi}{2} + x\right)$; b) $\cos\left(\dfrac{\pi}{2} + x\right)$; c) $\operatorname{sen}\left(\dfrac{3\pi}{2} - x\right)$; d) $\cos\left(\dfrac{3\pi}{2} - x\right)$; e) $\operatorname{sen}\left(\dfrac{3\pi}{2} + x\right)$; f) $\cos\left(\dfrac{3\pi}{2} + x\right)$.

See the resolution
$$\text{a)}\ \operatorname{sen}\left(\tfrac{\pi}{2} + x\right) = \operatorname{sen}\left[\pi - \left(\tfrac{\pi}{2} + x\right)\right] = \operatorname{sen}\left(\tfrac{\pi}{2} - x\right) = \cos x$$$$\text{b)}\ \cos\left(\tfrac{\pi}{2} + x\right) = -\cos\left[\pi - \left(\tfrac{\pi}{2} + x\right)\right] = -\cos\left(\tfrac{\pi}{2} - x\right) = -\operatorname{sen} x$$$$\text{c)}\ \operatorname{sen}\left(\tfrac{3\pi}{2} - x\right) = -\operatorname{sen}\left[\left(\tfrac{3\pi}{2} - x\right) - \pi\right] = -\operatorname{sen}\left(\tfrac{\pi}{2} - x\right) = -\cos x$$$$\text{d)}\ \cos\left(\tfrac{3\pi}{2} - x\right) = -\cos\left[\left(\tfrac{3\pi}{2} - x\right) - \pi\right] = -\cos\left(\tfrac{\pi}{2} - x\right) = -\operatorname{sen} x$$$$\text{e)}\ \operatorname{sen}\left(\tfrac{3\pi}{2} + x\right) = -\operatorname{sen}\left[2\pi - \left(\tfrac{3\pi}{2} + x\right)\right] = -\operatorname{sen}\left(\tfrac{\pi}{2} - x\right) = -\cos x$$$$\text{f)}\ \cos\left(\tfrac{3\pi}{2} + x\right) = \cos\left[2\pi - \left(\tfrac{3\pi}{2} + x\right)\right] = \cos\left(\tfrac{\pi}{2} - x\right) = \operatorname{sen} x$$
120

Simplify $y = \dfrac{\operatorname{sen}(2\pi - x) \cdot \cos(\pi - x)}{\operatorname{tg}\left(\frac{\pi}{2} + x\right) \cdot \operatorname{cotg}\left(\frac{3\pi}{2} - x\right)}$.

See the resolution

Using the reductions, $\operatorname{tg}\left(\frac{\pi}{2} + x\right) = \dfrac{\cos x}{-\operatorname{sen} x} = -\operatorname{cotg} x$ and $\operatorname{cotg}\left(\frac{3\pi}{2} - x\right) = \dfrac{-\operatorname{sen} x}{-\cos x} = \operatorname{tg} x$. Then

$$y = \frac{(-\operatorname{sen} x)(-\cos x)}{(-\operatorname{cotg} x)(\operatorname{tg} x)} = -\operatorname{sen} x \cdot \cos x.$$
121

Simplify the expressions:

$$\text{a)}\ \frac{\operatorname{sen}(-x) \cdot \cos\left(\frac{\pi}{2} + x\right)}{\operatorname{tg}(2\pi - x) \cdot \cos(\pi - x)} \qquad \text{b)}\ \frac{\operatorname{sen}(180^\circ - x) \cdot \operatorname{tg}(90^\circ + x)}{\operatorname{cotg}(270^\circ + x) \cdot \cos(270^\circ - x)}$$$$\text{c)}\ \frac{\sec(\pi - x) \cdot \operatorname{tg}\left(x - \frac{\pi}{2}\right)}{\operatorname{cossec}(9\pi - x) \cdot \operatorname{cotg}(-x)} \qquad \text{d)}\ \operatorname{sen}\left(\frac{3\pi}{2} - x\right) + \cos(4\pi - x) + \operatorname{tg}\left(\frac{3\pi}{2} - x\right)$$
Ver resposta

a) $\operatorname{sen} x$ · b) $-\operatorname{cotg}^2 x$ · c) $-\operatorname{tg} x$ · d) $\operatorname{cotg} x$

122

(MAPOFEI-76) Simplify the expression $\operatorname{sen}\dfrac{9\pi}{2} - \cos\left(x + \dfrac{15\pi}{2}\right) \cdot \operatorname{sen}(7\pi - x)$.

Ver resposta

$\cos^2 x$

123

(MAPOFEI-74) Simplify the expression $\operatorname{sen}\dfrac{7\pi}{2} + \dfrac{\operatorname{sen}(x + 11\pi) \cdot \operatorname{cotg}\left(x + \frac{11\pi}{2}\right)}{\cos(9\pi - x)}$.

Ver resposta

$-\sec^2 x$

124

(MAPOFEI-74) Simplify the expression $\dfrac{a^2 \cos 180^\circ - (a - b)^2 \operatorname{sen} 270^\circ + 2ab \cos 0^\circ}{b^2 \operatorname{sen} 90^\circ}$.

Ver resposta

$1$

125

(MAPOFEI-76) Graph the function $y = \operatorname{sen}\left(x - \dfrac{\pi}{2}\right) + 2$.

See the answer

As $\operatorname{sen}\left(x - \frac{\pi}{2}\right) = -\cos x$, we have $y = 2 - \cos x$: image $[1, 3]$ and period $2\pi$.

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