Vamos resolver a questão passo a passo, começando com a informação dada: \(\operatorname{sen} x = \frac{1}{2}\) e \(0 \leqslant x \leqslant \frac{\pi}{2}\). 1. Encontrando \(\cos x\): Sabemos que \(\sin^2 x + \cos^2 x = 1\). Portanto: \[ \left(\frac{1}{2}\right)^2 + \cos^2 x = 1 \] \[ \frac{1}{4} + \cos^2 x = 1 \] \[ \cos^2 x = 1 - \frac{1}{4} = \frac{3}{4} \] \[ \cos x = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \] 2. Calculando \(\cos \left(x + \frac{\pi}{2}\right)\): Usando a identidade \(\cos\left(x + \frac{\pi}{2}\right) = -\sin x\): \[ \cos\left(x + \frac{\pi}{2}\right) = -\sin x = -\frac{1}{2} \] 3. Calculando \(\sin\left(x + \frac{\pi}{2}\right)\): Usando a identidade \(\sin\left(x + \frac{\pi}{2}\right) = \cos x\): \[ \sin\left(x + \frac{\pi}{2}\right) = \cos x = \frac{\sqrt{3}}{2} \] 4. Calculando \(\tan\left(x + \frac{\pi}{2}\right)\): Usando a identidade \(\tan\left(x + \frac{\pi}{2}\right) = -\cot x\): \[ \tan\left(x + \frac{\pi}{2}\right) = -\cot x = -\frac{1}{\tan x} = -\frac{\cos x}{\sin x} = -\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3} \] 5. Calculando \(\cot\left(x + \frac{\pi}{2}\right)\): Usando a identidade \(\cot\left(x + \frac{\pi}{2}\right) = -\tan x\): \[ \cot\left(x + \frac{\pi}{2}\right) = -\tan x = -\frac{1}{\sqrt{3}} \] 6. Calculando \(\sec\left(x + \frac{\pi}{2}\right)\): Usando a identidade \(\sec\left(x + \frac{\pi}{2}\right) = -\csc x\): \[ \sec\left(x + \frac{\pi}{2}\right) = -\csc x = -\frac{1}{\sin x} = -2 \] 7. Calculando \(\csc\left(x + \frac{\pi}{2}\right)\): Usando a identidade \(\csc\left(x + \frac{\pi}{2}\right) = \sec x\): \[ \csc\left(x + \frac{\pi}{2}\right) = \sec x = \frac{1}{\cos x} = \frac{2}{\sqrt{3}} \] Resumindo: - a) \(\cos x = \frac{\sqrt{3}}{2}\) - b) \(\cos\left(x + \frac{\pi}{2}\right) = -\frac{1}{2}\) - c) \(\sin\left(x + \frac{\pi}{2}\right) = \frac{\sqrt{3}}{2}\) - d) \(\tan\left(x + \frac{\pi}{2}\right) = -\sqrt{3}\) - e) \(\cot\left(x + \frac{\pi}{2}\right) = -\frac{1}{\sqrt{3}}\) - f) \(\sec\left(x + \frac{\pi}{2}\right) = -2\) - g) \(\csc\left(x + \frac{\pi}{2}\right) = \frac{2}{\sqrt{3}}\) Se precisar de mais alguma coisa, é só avisar!