Question

1-49. δpat is an isosceles right triangle with pa = ta a. calculate the exact length of (overline{pa}). express your answer in simple radical form. b. what is the exact value of sin(p)? c. what is the exact value of cos(p)?

Explanation:

Step1: Apply Pythagorean theorem

In right - triangle $\triangle PAT$ with $PA = TA$ and $PT = 8$, by the Pythagorean theorem $PA^{2}+TA^{2}=PT^{2}$. Since $PA = TA$, we have $2PA^{2}=8^{2}=64$.

Step2: Solve for $PA$

Dividing both sides of $2PA^{2}=64$ by 2 gives $PA^{2}=32$. Then taking the square - root of both sides, $PA=\sqrt{32}=4\sqrt{2}$.

Step3: Calculate $\sin(P)$

In a right - triangle, $\sin(P)=\frac{TA}{PT}$. Since $TA = PA = 4\sqrt{2}$ and $PT = 8$, then $\sin(P)=\frac{4\sqrt{2}}{8}=\frac{\sqrt{2}}{2}$.

Step4: Calculate $\cos(P)$

In a right - triangle, $\cos(P)=\frac{PA}{PT}$. Since $PA = 4\sqrt{2}$ and $PT = 8$, then $\cos(P)=\frac{4\sqrt{2}}{8}=\frac{\sqrt{2}}{2}$.

Answer:

a. $4\sqrt{2}$
b. $\frac{\sqrt{2}}{2}$
c. $\frac{\sqrt{2}}{2}$