Explain the relationship between the sine and cosine of complementary angles and justify your reasoning,
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Explain the relationship between the sine and cosine of complementary angles and justify your reasoning, Drag the correct component into the boxes to correctly model the situation. Show Hints

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Certainly! Let's explore the relationship between the sine and cosine of complementary angles and understand why this relationship holds true.

### **Understanding Complementary Angles**

- **Complementary Angles**: Two angles are complementary if the sum of their measures is **90 degrees**. For example, if one angle is $$ \theta $$, its complement is $$ 90^\circ - \theta $$.

### **Relationship Between Sine and Cosine of Complementary Angles**

- **Key Relationship**:
  $$
  \sin(\theta) = \cos(90^\circ - \theta) \quad \text{and} \quad \cos(\theta) = \sin(90^\circ - \theta)
  $$

This means that the sine of an angle is equal to the cosine of its complement, and vice versa.

### **Justification Using a Right Triangle**

Consider a right triangle where one of the non-right angles is $$ \theta $$. Let's label the sides as follows:

- **Adjacent Side to $$ \theta $$**: $$ \text{adj} $$
- **Opposite Side to $$ \theta $$**: $$ \text{opp} $$
- **Hypotenuse**: $$ \text{hyp} $$

Since the two non-right angles in a right triangle are complementary ($$ \theta $$ and $$ 90^\circ - \theta $$), we can analyze the sine and cosine of these angles.

1. **Sine of $$ \theta $$**:
   $$
   \sin(\theta) = \frac{\text{opp}}{\text{hyp}}
   $$

2. **Cosine of $$ 90^\circ - \theta $$**:
   - Notice that $$ 90^\circ - \theta $$ is the other acute angle in the triangle.
   - The **adjacent side** to $$ \theta $$ is the **opposite side** to $$ 90^\circ - \theta $$.
   $$
   \cos(90^\circ - \theta) = \frac{\text{adj}}{\text{hyp}}
   $$

But in the context of replacing the sides:
$$
\cos(90^\circ - \theta) = \frac{\text{adj}}{\text{hyp}} = \frac{\text{opp}}{\text{hyp}} = \sin(\theta)
$$
Thus:
$$
\sin(\theta) = \cos(90^\circ - \theta)
$$

Similarly:
$$
\cos(\theta) = \sin(90^\circ - \theta)
$$

### **Visual Representation**

Imagine a right triangle where one acute angle is $$ \theta $$ and the other is $$ 90^\circ - \theta $$. The roles of the opposite and adjacent sides switch when you consider the complementary angle:

- For angle $$ \theta $$:
  - **Opposite**: $$ \text{opp} $$
  - **Adjacent**: $$ \text{adj} $$

- For angle $$ 90^\circ - \theta $$:
  - **Opposite**: $$ \text{adj} $$
  - **Adjacent**: $$ \text{opp} $$

This switch clearly shows why $$ \sin(\theta) $$ is equal to $$ \cos(90^\circ - \theta) $$ and vice versa.

### **Conclusion**

The sine of an angle is equal to the cosine of its complementary angle, and the cosine of an angle is equal to the sine of its complementary angle. This relationship stems from the definitions of sine and cosine in a right triangle and the fact that the non-right angles in a right triangle are complementary.
The sine of an angle is equal to the cosine of its complement, and the cosine of an angle is equal to the sine of its complement.

Explain the relationship between the sine and cosine of complementary angles and justify your reasoning,
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