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To understand why similar right triangles prove that the sine and cosine of corresponding angles are equal, let's follow a step-by-step explanation.
### Key Concepts:
1. Right Triangle:
- A triangle with one 90-degree right angle.
2. Similar Triangles:
- Triangles that have the same shape but different sizes. They have identical angle measures and their sides are proportional.
3. Sine and Cosine Definitions:
- For a given angle in a right triangle:
- Sine (sin) of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
### Explanation:
1. Consider Two Similar Right Triangles:
- Let's label the larger triangle [tex]\( \triangle ABC \)[/tex] and the smaller triangle [tex]\( \triangle DEF \)[/tex].
- Both triangles have a right angle.
- Let [tex]\( \angle A = \angle D \)[/tex] (corresponding angles).
2. Proportional Sides:
- Since [tex]\( \triangle ABC \sim \triangle DEF \)[/tex], the sides of these triangles are proportional. That is, the ratios of the lengths of corresponding sides are equal.
- Specifically,
[tex]\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \][/tex]
3. Simplifying the Ratios:
- Let [tex]\( \angle A \)[/tex] in [tex]\( \triangle ABC \)[/tex] be the same as [tex]\( \angle D \)[/tex] in [tex]\( \triangle DEF \)[/tex].
- Let’s analyze the sine (sin) and cosine (cos) for these angles:
- In [tex]\( \triangle ABC \)[/tex]:
- [tex]\( \sin(\angle A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} \)[/tex]
- [tex]\( \cos(\angle A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} \)[/tex]
- In [tex]\( \triangle DEF \)[/tex]:
- [tex]\( \sin(\angle D) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{EF}{DF} \)[/tex]
- [tex]\( \cos(\angle D) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{DE}{DF} \)[/tex]
4. Using the Proportional Sides Property:
- As per the similarity of triangles, we have:
[tex]\[ \frac{BC}{AC} = \frac{EF}{DF} \quad \text{(sin ratios are equal)} \][/tex]
[tex]\[ \frac{AB}{AC} = \frac{DE}{DF} \quad \text{(cos ratios are equal)} \][/tex]
- This indicates that:
- [tex]\( \sin(\angle A) = \sin(\angle D) \)[/tex]
- [tex]\( \cos(\angle A) = \cos(\angle D) \)[/tex]
### Conclusion:
By the properties of similar triangles, the ratios of corresponding sides are equal. For a given angle in similar right triangles, the opposite side-to-hypotenuse ratio (sine) and the adjacent side-to-hypotenuse ratio (cosine) remain constant. Thus, similar right triangles confirm that the sine and cosine functions for corresponding angles are equal.
This fundamental property is a cornerstone of trigonometric functions and their relation to angular measures in right triangles.
### Key Concepts:
1. Right Triangle:
- A triangle with one 90-degree right angle.
2. Similar Triangles:
- Triangles that have the same shape but different sizes. They have identical angle measures and their sides are proportional.
3. Sine and Cosine Definitions:
- For a given angle in a right triangle:
- Sine (sin) of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
### Explanation:
1. Consider Two Similar Right Triangles:
- Let's label the larger triangle [tex]\( \triangle ABC \)[/tex] and the smaller triangle [tex]\( \triangle DEF \)[/tex].
- Both triangles have a right angle.
- Let [tex]\( \angle A = \angle D \)[/tex] (corresponding angles).
2. Proportional Sides:
- Since [tex]\( \triangle ABC \sim \triangle DEF \)[/tex], the sides of these triangles are proportional. That is, the ratios of the lengths of corresponding sides are equal.
- Specifically,
[tex]\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \][/tex]
3. Simplifying the Ratios:
- Let [tex]\( \angle A \)[/tex] in [tex]\( \triangle ABC \)[/tex] be the same as [tex]\( \angle D \)[/tex] in [tex]\( \triangle DEF \)[/tex].
- Let’s analyze the sine (sin) and cosine (cos) for these angles:
- In [tex]\( \triangle ABC \)[/tex]:
- [tex]\( \sin(\angle A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} \)[/tex]
- [tex]\( \cos(\angle A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} \)[/tex]
- In [tex]\( \triangle DEF \)[/tex]:
- [tex]\( \sin(\angle D) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{EF}{DF} \)[/tex]
- [tex]\( \cos(\angle D) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{DE}{DF} \)[/tex]
4. Using the Proportional Sides Property:
- As per the similarity of triangles, we have:
[tex]\[ \frac{BC}{AC} = \frac{EF}{DF} \quad \text{(sin ratios are equal)} \][/tex]
[tex]\[ \frac{AB}{AC} = \frac{DE}{DF} \quad \text{(cos ratios are equal)} \][/tex]
- This indicates that:
- [tex]\( \sin(\angle A) = \sin(\angle D) \)[/tex]
- [tex]\( \cos(\angle A) = \cos(\angle D) \)[/tex]
### Conclusion:
By the properties of similar triangles, the ratios of corresponding sides are equal. For a given angle in similar right triangles, the opposite side-to-hypotenuse ratio (sine) and the adjacent side-to-hypotenuse ratio (cosine) remain constant. Thus, similar right triangles confirm that the sine and cosine functions for corresponding angles are equal.
This fundamental property is a cornerstone of trigonometric functions and their relation to angular measures in right triangles.
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