Fill in the blank to complete the fundamental identity.
[tex]$\sin ^2 \theta+\cos ^2 \theta=$[/tex]
Answer (2)
Recall the Pythagorean trigonometric identity.
State the identity: sin 2 θ + cos 2 θ = 1 .
Complete the given expression using the identity.
The completed fundamental identity is: 1 .
Explanation
Problem Analysis The problem asks us to complete a fundamental trigonometric identity. We need to recall the Pythagorean identity, which relates the sine and cosine of an angle.
Recall Pythagorean Identity The Pythagorean identity states that for any angle θ , the sum of the square of the sine of θ and the square of the cosine of θ is equal to 1.
State the Completed Identity Therefore, the completed identity is: sin 2 θ + cos 2 θ = 1
Examples
In construction and architecture, when calculating angles and forces in structures, the Pythagorean trigonometric identity is essential. For example, when designing a roof, engineers use trigonometric functions to determine the correct angles for stability and water runoff. The identity sin 2 θ + cos 2 θ = 1 helps simplify these calculations, ensuring the roof is both functional and safe.
The fundamental identity sin 2 θ + cos 2 θ completes to 1, according to the Pythagorean trigonometric identity. This identity is essential in trigonometry and is used in various practical applications, including engineering and architecture. Thus, the completed identity is: sin 2 θ + cos 2 θ = 1 .
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