Question

move expressions and reasons into the blanks to complete the proof of sin²θ + cos²θ = 1. according to _____________, it is known that a² + b² = 1. by the definition of sine, sinθ = . by the definition of cosine, cosθ = . therefore by _____________, sin²θ + cos²θ = 1 the pythagorean identity the pythagorean theorem combining like terms substitution a 1/a a/b b/a b 1/b

Explanation:

Step1: Recall Pythagorean Theorem

In a right - triangle formed in the unit circle with sides \(a\), \(b\) and hypotenuse \(1\), by the Pythagorean Theorem \(a^{2}+b^{2}=1\).

Step2: Define sine

By the definition of sine in a right - triangle in the unit circle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} = a\).

Step3: Define cosine

By the definition of cosine in a right - triangle in the unit circle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=b\).

Step4: Substitute

Substitute \(\sin\theta = a\) and \(\cos\theta = b\) into \(\sin^{2}\theta+\cos^{2}\theta\). We get \(\sin^{2}\theta+\cos^{2}\theta=a^{2}+b^{2}\). Since \(a^{2}+b^{2}=1\), by substitution \(\sin^{2}\theta+\cos^{2}\theta = 1\).

Answer:

1. the Pythagorean Theorem
2. \(a\)
3. \(b\)
4. substitution