Given the Pythagorean theorem a² + b² = c² where a and b are the lengths of the perpendicular sides of a right triangle and c is the length of the hypotenuse. Derive the trigonometric identity sin² (0) + cos² (0) = 1.
Given the Pythagorean theorem a² + b² = c² where a and b are the lengths of the perpendicular sides of a right triangle and c is the length of the hypotenuse. Derive the trigonometric identity sin² (0) + cos² (0) = 1.
Calculus: Early Transcendentals
8th Edition
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Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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![Given the Pythagorean theorem
\[ a^2 + b^2 = c^2 \]
where \( a \) and \( b \) are the lengths of the perpendicular sides of a right triangle and \( c \) is the length of the hypotenuse. Derive the trigonometric identity
\[ \sin^2(\theta) + \cos^2(\theta) = 1. \]
Explanation:
1. **Pythagorean Theorem**: The theorem states that in a right triangle, the square of the length of the hypotenuse \( c \) is equal to the sum of the squares of the lengths of the other two sides \( a \) and \( b \).
2. **Right Triangle Definitions**:
- Sine of angle \( \theta \): \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
- Cosine of angle \( \theta \): \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
3. **Expressing in terms of a, b, and c**:
- Let us assume that in the given right triangle, side \( a \) is opposite to angle \( \theta \), side \( b \) is adjacent to angle \( \theta \), and \( c \) is the hypotenuse.
- Then \(\sin(\theta) = \frac{a}{c}\) and \(\cos(\theta) = \frac{b}{c}\).
4. **Square both sides**:
- \(\sin^2(\theta) = \left(\frac{a}{c}\right)^2 = \frac{a^2}{c^2}\)
- \(\cos^2(\theta) = \left(\frac{b}{c}\right)^2 = \frac{b^2}{c^2}\)
5. **Add the squared terms**:
- \(\sin^2(\theta) + \cos^2(\theta) = \frac{a^2}{c^2} + \frac{b^2}{c^2} = \frac{a^2 + b^2}{c^2}\)
6. **Apply the Pythagorean theorem**:
- According to the Pythagorean theorem, \( a^2 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1d592a5-ec96-4b1d-bddf-23e27014debe%2Fad3f891d-1f67-43af-8ad3-fce80b707028%2Fch0vlpg_processed.png&w=3840&q=75)
Transcribed Image Text:Given the Pythagorean theorem
\[ a^2 + b^2 = c^2 \]
where \( a \) and \( b \) are the lengths of the perpendicular sides of a right triangle and \( c \) is the length of the hypotenuse. Derive the trigonometric identity
\[ \sin^2(\theta) + \cos^2(\theta) = 1. \]
Explanation:
1. **Pythagorean Theorem**: The theorem states that in a right triangle, the square of the length of the hypotenuse \( c \) is equal to the sum of the squares of the lengths of the other two sides \( a \) and \( b \).
2. **Right Triangle Definitions**:
- Sine of angle \( \theta \): \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
- Cosine of angle \( \theta \): \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
3. **Expressing in terms of a, b, and c**:
- Let us assume that in the given right triangle, side \( a \) is opposite to angle \( \theta \), side \( b \) is adjacent to angle \( \theta \), and \( c \) is the hypotenuse.
- Then \(\sin(\theta) = \frac{a}{c}\) and \(\cos(\theta) = \frac{b}{c}\).
4. **Square both sides**:
- \(\sin^2(\theta) = \left(\frac{a}{c}\right)^2 = \frac{a^2}{c^2}\)
- \(\cos^2(\theta) = \left(\frac{b}{c}\right)^2 = \frac{b^2}{c^2}\)
5. **Add the squared terms**:
- \(\sin^2(\theta) + \cos^2(\theta) = \frac{a^2}{c^2} + \frac{b^2}{c^2} = \frac{a^2 + b^2}{c^2}\)
6. **Apply the Pythagorean theorem**:
- According to the Pythagorean theorem, \( a^2 +
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