1. Sendo cos x = 4/5 e 0

Question

1. Sendo cos x = 4/5 e 0 < x < π/2, calcule o valor de sen²x - 3.sen x.

2. Sabendo que cos a = -√5/5 e π/2 < a < π, determine o valor de (1 + sen a)(1 - sen a).

3. Dado sen x = 1/3, com π/2 < x < π, determine o valor da cotg x.

4. Dado cos x = √2/2, com 0 < x < π/2, determine o valor de sec x + cos x.

5. Sendo sec x = 7/3, determine o valor de cotg x.

Accepted Answer

Vamos resolver os problemas um por um:

1. **Sendo $\cos x = \frac{4}{5}$ e $0 < x < \frac{\pi}{2}$, calcule o valor de $\sen^2 x - 3 \cdot \sen x$.**

Primeiro, calcule $\sen x$ usando a identidade $\sen^2 x + \cos^2 x = 1$:

$$
   \sen^2 x = 1 - \cos^2 x = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25}
   $$

Portanto, $\sen x = \frac{3}{5}$.

Agora, calcule $\sen^2 x - 3 \cdot \sen x$:

$$
   \sen^2 x - 3 \cdot \sen x = \frac{9}{25} - 3 \cdot \frac{3}{5} = \frac{9}{25} - \frac{9}{5} = \frac{9}{25} - \frac{45}{25} = -\frac{36}{25}
   $$

2. **Sabendo que $\cos a = -\frac{\sqrt{15}}{5}$ e $\frac{\pi}{2} < a < \pi$, determine o valor de $(1 + \sen a)(1 - \sen a)$.**

Primeiro, calcule $\sen a$ usando $\sen^2 a + \cos^2 a = 1$:

$$
   \sen^2 a = 1 - \left(-\frac{\sqrt{15}}{5}\right)^2 = 1 - \frac{15}{25} = \frac{10}{25} = \frac{2}{5}
   $$

Como $\frac{\pi}{2} < a < \pi$, $\sen a$ é positivo: $\sen a = \sqrt{\frac{2}{5}}$.

Calcule $(1 + \sen a)(1 - \sen a)$:

$$
   (1 + \sen a)(1 - \sen a) = 1 - (\sen a)^2 = 1 - \frac{2}{5} = \frac{3}{5}
   $$

3. **Dado $\sen x = \frac{1}{3}$, com $\frac{\pi}{2} < x < \pi$, determine o valor da $\cot x$.**

Calcule $\cos x$ usando $\sen^2 x + \cos^2 x = 1$:

$$
   \cos^2 x = 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}
   $$

Como $\frac{\pi}{2} < x < \pi$, $\cos x$ é negativo: $\cos x = -\sqrt{\frac{8}{9}} = -\frac{2\sqrt{2}}{3}$.

Calcule $\cot x = \frac{\cos x}{\sen x}$:

$$
   \cot x = \frac{-\frac{2\sqrt{2}}{3}}{\frac{1}{3}} = -2\sqrt{2}
   $$

4. **Dado $\cos x = \frac{\sqrt{2}}{2}$, com $0 < x < \frac{\pi}{2}$, determine o valor de $\sec x + \csc x$.**

Sabemos que $\cos x = \frac{\sqrt{2}}{2}$ implica $\sec x = \frac{1}{\cos x} = \sqrt{2}$.

Calcule $\sen x = \sqrt{1 - \cos^2 x} = \sqrt{1 - \left(\frac{\sqrt{2}}{2}\right)^2} = \frac{\sqrt{2}}{2}$.

Portanto, $\csc x = \frac{1}{\sen x} = \sqrt{2}$.

Assim, $\sec x + \csc x = \sqrt{2} + \sqrt{2} = 2\sqrt{2}$.

5. **Sendo $\sec x = \frac{7}{3}$, determine o valor de $\cot x$.**

Sabemos que $\sec x = \frac{1}{\cos x}$, então $\cos x = \frac{3}{7}$.

Calcule $\sen x$ usando $\sen^2 x + \cos^2 x = 1$:

$$
   \sen^2 x = 1 - \left(\frac{3}{7}\right)^2 = 1 - \frac{9}{49} = \frac{40}{49}
   $$

Portanto, $\sen x = \frac{\sqrt{40}}{7} = \frac{2\sqrt{10}}{7}$.

Calcule $\cot x = \frac{\cos x}{\sen x}$:

$$
   \cot x = \frac{\frac{3}{7}}{\frac{2\sqrt{10}}{7}} = \frac{3}{2\sqrt{10}} = \frac{3\sqrt{10}}{20}
   $$

1. Sendo cos x = 4/5 e 0 < x < π/2, calcule o valor de sen²x...

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