https://r2.erweima.ai/mathbot/image/1726535171446ndbs4mit.jpg

Let's analyze the two questions from the image and solve them one by one.

### 1. **Calcula el valor de las razones trigonométricas para los ángulos que se presentan a continuación:**
   - \( a. \, 0 \, \text{rad} \)
   - \( b. \, \frac{\pi}{2} \, \text{rad} \)
   - \( c. \, \pi \, \text{rad} \)
   - \( d. \, \frac{3\pi}{2} \, \text{rad} \)

For each of these angles, we calculate the sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)):

- **(a) \( 0 \, \text{rad} \)**:
   - \( \sin(0) = 0 \)
   - \( \cos(0) = 1 \)
   - \( \tan(0) = 0 \)

- **(b) \( \frac{\pi}{2} \, \text{rad} \)**:
   - \( \sin\left(\frac{\pi}{2}\right) = 1 \)
   - \( \cos\left(\frac{\pi}{2}\right) = 0 \)
   - \( \tan\left(\frac{\pi}{2}\right) \) is undefined because division by zero occurs when \( \cos = 0 \).

- **(c) \( \pi \, \text{rad} \)**:
   - \( \sin(\pi) = 0 \)
   - \( \cos(\pi) = -1 \)
   - \( \tan(\pi) = 0 \)

- **(d) \( \frac{3\pi}{2} \, \text{rad} \)**:
   - \( \sin\left(\frac{3\pi}{2}\right) = -1 \)
   - \( \cos\left(\frac{3\pi}{2}\right) = 0 \)
   - \( \tan\left(\frac{3\pi}{2}\right) \) is undefined because \( \cos = 0 \).

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### 2. **Indica el signo de todas las razones trigonométricas de los siguientes ángulos expresados en grados:**

Here, we determine the sign of sine, cosine, and tangent for each angle:

- **(a) \( 120^\circ \)**: Quadrant II
   - \( \sin(120^\circ) > 0 \)
   - \( \cos(120^\circ) < 0 \)
   - \( \tan(120^\circ) < 0 \)

- **(b) \( -70^\circ \)**: Quadrant IV
   - \( \sin(-70^\circ) < 0 \)
   - \( \cos(-70^\circ) > 0 \)
   - \( \tan(-70^\circ) < 0 \)

- **(c) \( 256^\circ \)**: Quadrant III
   - \( \sin(256^\circ) < 0 \)
   - \( \cos(256^\circ) < 0 \)
   - \( \tan(256^\circ) > 0 \)

- **(d) \( 800^\circ \)**: Equivalent to \( 800^\circ - 2 \times 360^\circ = 80^\circ \) in Quadrant I
   - \( \sin(80^\circ) > 0 \)
   - \( \cos(80^\circ) > 0 \)
   - \( \tan(80^\circ) > 0 \)

- **(e) \( 315^\circ \)**: Quadrant IV
   - \( \sin(315^\circ) < 0 \)
   - \( \cos(315^\circ) > 0 \)
   - \( \tan(315^\circ) < 0 \)

- **(f) \( 1200^\circ \)**: Equivalent to \( 1200^\circ - 3 \times 360^\circ = 120^\circ \) in Quadrant II
   - \( \sin(120^\circ) > 0 \)
   - \( \cos(120^\circ) < 0 \)
   - \( \tan(120^\circ) < 0 \)

- **(g) \( 55^\circ \)**: Quadrant I
   - \( \sin(55^\circ) > 0 \)
   - \( \cos(55^\circ) > 0 \)
   - \( \tan(55^\circ) > 0 \)

- **(h) \( -460^\circ \)**: Equivalent to \( -460^\circ + 360^\circ = -100^\circ \) in Quadrant III
   - \( \sin(-100^\circ) < 0 \)
   - \( \cos(-100^\circ) < 0 \)
   - \( \tan(-100^\circ) > 0 \)

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Would you like further details or explanations on any part of these solutions?

Here are 5 related questions to practice:
1. What are the exact trigonometric values for \( \frac{\pi}{3} \, \text{rad} \)?
2. How do you convert \( 210^\circ \) into radians?
3. What is the cosine of an angle in Quadrant III if the tangent is positive?
4. Determine the sine, cosine, and tangent of \( 450^\circ \).
5. What quadrant does the angle \( -150^\circ \) lie in, and what are the signs of its trigonometric functions?

**Tip:** To determine the reference angle for large angles, always subtract or add \( 360^\circ \) (or \( 2\pi \) radians) until you obtain an angle between \( 0^\circ \) and \( 360^\circ \).

Calcula el valor de las razones trigonométricas para los ángulos que se presentan a continuación: 0 rad, π/2 rad, π rad, 3π/2 rad. Indica el signo de todas las razones trigonométricas de los siguientes ángulos expresados en grados: 120°, -70°, 256°, 800°, 315°, 1200°, 55°, -460°.

Explore how to calculate trigonometric values for angles such as 0 rad, π/2 rad, π rad, and 3π/2 rad, and determine the signs of trigonometric functions for angles in degrees including 120°, -70°, and more. Suitable for high school students studying trigonometry.

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