Match the trigonometric functions that are related through a pythagorean identity.
The Correct Answer and Explanation is :
Pythagorean identities are fundamental relationships in trigonometry derived from the Pythagorean theorem. The primary trigonometric functions—sine (sin), cosine (cos), and tangent (tan)—exhibit these identities, which can be expressed in various forms. Here, we will focus on the primary Pythagorean identity:
[
\sin^2(x) + \cos^2(x) = 1
]
Related Trigonometric Functions
- Sine and Cosine:
- From the identity (\sin^2(x) + \cos^2(x) = 1), we can derive relationships involving sine and cosine directly. This identity shows that for any angle (x), the square of the sine of that angle plus the square of the cosine equals one.
- Tangent and Secant:
- The relationship between tangent and secant can be derived from the first identity. Recall that:
[
\tan(x) = \frac{\sin(x)}{\cos(x)}
]
If we divide the entire Pythagorean identity by (\cos^2(x)), we obtain:
[
\tan^2(x) + 1 = \sec^2(x)
] - Thus, we have another Pythagorean identity:
[
\tan^2(x) + 1 = \sec^2(x)
]
- Cotangent and Cosecant:
- Similarly, we can establish a relationship between cotangent and cosecant. Dividing the first identity by (\sin^2(x)) gives:
[
1 + \cot^2(x) = \csc^2(x)
] - Therefore, this yields the third Pythagorean identity:
[
1 + \cot^2(x) = \csc^2(x)
]
Summary
In summary, the three primary Pythagorean identities are:
- (\sin^2(x) + \cos^2(x) = 1)
- (\tan^2(x) + 1 = \sec^2(x))
- (1 + \cot^2(x) = \csc^2(x))
These identities reflect the relationships between the basic trigonometric functions and are instrumental in solving various trigonometric equations and simplifying expressions in mathematics. Understanding these identities is crucial for anyone studying trigonometry, as they provide a foundation for more complex concepts and applications.