Question

given: △abc is a right triangle. prove: (sin(a))² + (cos(a))² = 1. drag reasons into the table to complete the proof. statement: △abc is a right triangle; reason: given. sin(a)=a/c; cos(a)=b/c; a² + b² = c²; a²/c² + b²/c² = 1; (a/c)²+(b/c)² = 1; (sin(a))² + (cos(a))² = 1. definition of cosine; definition of sine; definition of right triangle; pythagorean theorem; exponent rules for division; substitution property; multiplication property of equality; transitive property of equality

Explanation:

Step1: Recall sine definition

By the definition of sine in a right - triangle, $\sin(A)=\frac{a}{c}$. Reason: Definition of sine.

Step2: Recall cosine definition

By the definition of cosine in a right - triangle, $\cos(A)=\frac{b}{c}$. Reason: Definition of cosine.

Step3: Apply Pythagorean theorem

In right - triangle $\triangle ABC$, $a^{2}+b^{2}=c^{2}$ by the Pythagorean theorem. Reason: Pythagorean Theorem.

Step4: Divide both sides by $c^{2}$

Dividing $a^{2}+b^{2}=c^{2}$ by $c^{2}$ gives $\frac{a^{2}}{c^{2}}+\frac{b^{2}}{c^{2}} = 1$. Reason: Division property of equality (equivalent to multiplying both sides by $\frac{1}{c^{2}}$).

Step5: Rewrite using exponent rules

We can rewrite $\frac{a^{2}}{c^{2}}$ as $(\frac{a}{c})^{2}$ and $\frac{b^{2}}{c^{2}}$ as $(\frac{b}{c})^{2}$, so $(\frac{a}{c})^{2}+(\frac{b}{c})^{2}=1$. Reason: Exponent rules for division.

Step6: Substitute sine and cosine

Substituting $\sin(A)=\frac{a}{c}$ and $\cos(A)=\frac{b}{c}$ into $(\frac{a}{c})^{2}+(\frac{b}{c})^{2}=1$ gives $(\sin(A))^{2}+(\cos(A))^{2}=1$. Reason: Substitution Property.

Answer:

Statement	Reason
$\sin(A)=\frac{a}{c}$	Definition of sine
$\cos(A)=\frac{b}{c}$	Definition of cosine
$a^{2}+b^{2}=c^{2}$	Pythagorean Theorem
$\frac{a^{2}}{c^{2}}+\frac{b^{2}}{c^{2}} = 1$	Division property of equality
$(\frac{a}{c})^{2}+(\frac{b}{c})^{2}=1$	Exponent rules for division
$(\sin(A))^{2}+(\cos(A))^{2}=1$	Substitution Property