Angle K is between 0° and 180°. Which equation results in two different values for angle K? Question 14 options:
Sin K = 0.58
tan K= -0.86
cos K = -0.72
Cos K = 0.67

For an angle \( K \) between \( 0^\circ \) and \( 180^\circ \), we need to determine which of the given trigonometric equations could have two different values.

1. **\(\sin K = 0.58\):**
   - The sine function is positive in both the first and second quadrants. Therefore, there are two angles within \( 0^\circ \) to \( 180^\circ \) that satisfy this equation: one in the first quadrant and one in the second quadrant. Thus, this equation **can have two values**.

2. **\(\tan K = -0.86\):**
   - The tangent function is negative in the second quadrant for angles between \( 0^\circ \) and \( 180^\circ \). However, the tangent function only crosses a single value once within this interval. Hence, this equation results in **one value**.

3. **\(\cos K = -0.72\):**
   - The cosine function is negative in the second quadrant. Within the interval \( 0^\circ \) to \( 180^\circ \), the cosine function will only have one value in the second quadrant. Therefore, this equation results in **one value**.

4. **\(\cos K = 0.67\):**
   - The cosine function is positive in the first quadrant. Within the interval \( 0^\circ \) to \( 180^\circ \), the cosine function will have one value in the first quadrant. So, this equation results in **one value**.

### Conclusion:
The equation that results in **two different values** for angle \( K \) between \( 0^\circ \) and \( 180^\circ \) is:
- **\(\sin K = 0.58\).**

Would you like more details or have any questions about this problem?

Here are 5 related questions to expand on this topic:
1. How can we find both angles for \(\sin K = 0.58\) using a calculator?
2. Why does the sine function have two positive values within \( 0^\circ \) to \( 180^\circ \)?
3. How many values can an equation like \(\cos K = x\) have for \( K \) between \( 0^\circ \) and \( 360^\circ \)?
4. What are the general properties of the sine, cosine, and tangent functions concerning their ranges and signs?
5. How would the solutions differ if we considered angles \( K \) between \( 0^\circ \) and \( 360^\circ \)?

**Tip:** Remember that the sine function is symmetric about \( 90^\circ \), which is why it has two values between \( 0^\circ \) and \( 180^\circ \) when positive.

Angle K is between 0° and 180°. Which equation results in two different values for angle K? Question 14 options: Sin K = 0.58 tan K= -0.86 cos K = -0.72 Cos K = 0.67

Explore which trigonometric equation results in two values for angle K between 0° and 180°. Learn about the sine, cosine, and tangent functions, and understand their behavior in different quadrants. Perfect for students in Grades 9-11.

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