Question: 01 Calcular: (sec²45° + tan45°) . cot37° − 2cos60° A)0 B)1 C)2 D)3 E)4 02 Calcular: (tan60° + sec30° − sen60°) sec60° A) 25/12 B) 25/24 C) 49/12 D) 49/24 E) 7/18 03 Calcular: (tan37° + csc53°) / (sec45° + csc45°) A) √2 B) 2√2 C) √2/2 D) √2/4 E) 1 04 Determine el valor de “x” en: 5x.sen53° − sec60° = 2x + (tan60° + cot30°)² A)3 B)5 C)7 D)9 E)11 05 Calcular: E = √(a+b+c) Siendo: a = tan37° + csc53° b = sen45° . sec45° c = 2sen60°, tan30°

1. **Problema 01:** Calcular $$ (\sec^2 45^\circ + \tan 45^\circ) \cdot \cot 37^\circ - 2 \cos 60^\circ $$ 2. **Fórmulas y valores importantes:** - $$\sec 45^\circ = \frac{1}{\cos 45^\circ} = \sqrt{2}$$ - $$\tan 45^\circ = 1$$ - $$\cot 37^\circ = \tan (90^\circ - 37^\circ) = \tan 53^\circ$$ (aproximadamente 1.327) - $$\cos 60^\circ = \frac{1}{2}$$ 3. **Cálculo paso a paso:** - $$\sec^2 45^\circ = (\sqrt{2})^2 = 2$$ - $$\sec^2 45^\circ + \tan 45^\circ = 2 + 1 = 3$$ - Multiplicamos por $$\cot 37^\circ$$: $$3 \times 1.327 = 3.981$$ - Restamos $$2 \cos 60^\circ = 2 \times \frac{1}{2} = 1$$: $$3.981 - 1 = 2.981 \approx 3$$ 4. **Respuesta:** La opción correcta es D) 3 --- 1. **Problema 02:** Calcular $$ (\tan 60^\circ + \sec 30^\circ - \sin 60^\circ) \sec 60^\circ $$ 2. **Valores trigonométricos:** - $$\tan 60^\circ = \sqrt{3} \approx 1.732$$ - $$\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$$ - $$\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$$ - $$\sec 60^\circ = \frac{1}{\cos 60^\circ} = 2$$ 3. **Cálculo:** - $$\tan 60^\circ + \sec 30^\circ - \sin 60^\circ = 1.732 + \frac{2}{\sqrt{3}} - 0.866$$ - Convertimos $$\frac{2}{\sqrt{3}}$$ a decimal: $$\approx 1.155$$ - Suma: $$1.732 + 1.155 - 0.866 = 2.021$$ - Multiplicamos por $$\sec 60^\circ = 2$$: $$2.021 \times 2 = 4.042$$ 4. **Convertimos a fracción exacta:** - $$\tan 60^\circ = \sqrt{3}$$ - $$\sec 30^\circ = \frac{2}{\sqrt{3}}$$ - $$\sin 60^\circ = \frac{\sqrt{3}}{2}$$ - Sumamos: $$\sqrt{3} + \frac{2}{\sqrt{3}} - \frac{\sqrt{3}}{2} = \frac{2\sqrt{3}}{2} + \frac{2}{\sqrt{3}} - \frac{\sqrt{3}}{2} = \frac{2\sqrt{3} - \sqrt{3}}{2} + \frac{2}{\sqrt{3}} = \frac{\sqrt{3}}{2} + \frac{2}{\sqrt{3}}$$ - Combinamos en común denominador $$2\sqrt{3}$$: $$\frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2\sqrt{3}}$$ $$\frac{2}{\sqrt{3}} = \frac{4}{2\sqrt{3}}$$ - Suma: $$\frac{3\sqrt{3} + 4}{2\sqrt{3}}$$ - Multiplicamos por $$\sec 60^\circ = 2$$: $$2 \times \frac{3\sqrt{3} + 4}{2\sqrt{3}} = \frac{3\sqrt{3} + 4}{\sqrt{3}}$$ - Racionalizamos: $$\frac{3\sqrt{3} + 4}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3 \times 3 + 4 \sqrt{3}}{3} = \frac{9 + 4 \sqrt{3}}{3}$$ - Aproximadamente $$\approx 4.04$$, que corresponde a la opción C) 49/12 (ya que $$\frac{49}{12} \approx 4.083$$) 5. **Respuesta:** C) 49/12 --- 1. **Problema 03:** Calcular $$ \frac{\tan 37^\circ + \csc 53^\circ}{\sec 45^\circ + \csc 45^\circ} $$ 2. **Valores trigonométricos:** - $$\tan 37^\circ \approx 0.753$$ - $$\csc 53^\circ = \frac{1}{\sin 53^\circ} \approx \frac{1}{0.799} = 1.251$$ - $$\sec 45^\circ = \sqrt{2} \approx 1.414$$ - $$\csc 45^\circ = \frac{1}{\sin 45^\circ} = \sqrt{2} \approx 1.414$$ 3. **Cálculo:** - Numerador: $$0.753 + 1.251 = 2.004$$ - Denominador: $$1.414 + 1.414 = 2.828$$ - División: $$\frac{2.004}{2.828} \approx 0.708$$ 4. **Comparación con opciones:** - $$\frac{\sqrt{2}}{2} = \frac{1.414}{2} = 0.707$$ 5. **Respuesta:** C) $$\frac{\sqrt{2}}{2}$$ --- 1. **Problema 04:** Determinar $$x$$ en: $$5x \sin 53^\circ - \sec 60^\circ = 2x + (\tan 60^\circ + \cot 30^\circ)^2$$ 2. **Valores trigonométricos:** - $$\sin 53^\circ \approx 0.799$$ - $$\sec 60^\circ = 2$$ - $$\tan 60^\circ = \sqrt{3} \approx 1.732$$ - $$\cot 30^\circ = \sqrt{3} \approx 1.732$$ 3. **Calculamos el cuadrado:** - $$\tan 60^\circ + \cot 30^\circ = 1.732 + 1.732 = 3.464$$ - $$ (3.464)^2 = 12$$ 4. **Sustituimos:** - $$5x \times 0.799 - 2 = 2x + 12$$ - $$3.995x - 2 = 2x + 12$$ 5. **Despejamos $$x$$:** - $$3.995x - 2x = 12 + 2$$ - $$1.995x = 14$$ - $$x = \frac{14}{1.995} \approx 7.02$$ 6. **Respuesta:** C) 7 --- 1. **Problema 05:** Calcular $$E = \sqrt{a + b + c}$$ con: - $$a = \tan 37^\circ + \csc 53^\circ$$ - $$b = \sin 45^\circ \cdot \sec 45^\circ$$ - $$c = 2 \sin 60^\circ \cdot \tan 30^\circ$$ 2. **Valores trigonométricos:** - $$\tan 37^\circ \approx 0.753$$ - $$\csc 53^\circ \approx 1.251$$ - $$\sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.707$$ - $$\sec 45^\circ = \sqrt{2} \approx 1.414$$ - $$\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$$ - $$\tan 30^\circ = \frac{1}{\sqrt{3}} \approx 0.577$$ 3. **Calculamos cada término:** - $$a = 0.753 + 1.251 = 2.004$$ - $$b = 0.707 \times 1.414 = 1.000$$ - $$c = 2 \times 0.866 \times 0.577 = 1.000$$ 4. **Sumamos:** - $$a + b + c = 2.004 + 1.000 + 1.000 = 4.004$$ 5. **Calculamos $$E$$:** - $$E = \sqrt{4.004} \approx 2$$ 6. **Respuesta:** $$E \approx 2$$