Question

Given the point (3, 4), which of the following shows that sinxcscx = 1?

Answer 1

$\bf sin(\theta)=\cfrac{opposite}{hypotenuse}\qquad csc(\theta)=\cfrac{hypotenuse}{opposite}\\\\ -------------------------------\\\\ \begin{array}{clclll} 3&,&4\\ \uparrow &&\uparrow \\ x&&y\\ a&&b\\ adjacent&&opposite\\ side&&side \end{array}\qquad \begin{array}{llll} \textit{using the pythagorean theorem} \\\\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \end{array} \\\\\\ hypotenuse\implies c=\sqrt{3^2+4^2}\implies c=\sqrt{25}\implies \boxed{c=5}\\\\ -------------------------------$

$\bf sin(x)=\cfrac{4}{5}\qquad csc(x)=\cfrac{5}{4}\implies sin(x)csc(x)\implies \cfrac{4}{5}\cdot \cfrac{5}{4}\implies 1$

Answer 2

Answer:

The correct option is (4/5)·(5/4) = 1.

Step-by-step explanation:

First you have to consider the point (3,4) in a graph, and think in the triangle defined by this point and the point (0,0) and (3,0) as you will see in the attachment.

When you look at that triangle and you evaluate it in terms of Pitagoras´s theorem, you see that one of its sides (called a) is 3 and the other (called b) is 4. The when you want to find the value of hypothenuse (h)

h² = a² + b² ⇒√h² = √(a² + b²) ⇒ h = √(3² + 4²) ⇒h= √25 ⇒ h = 5

With this information and keeping in mind the definitions of sin(x) and csc(x)

sin(x) = b/h and csc(x) = 1/sin(x)

⇒ sin(x) = 4/5.

⇒ csc(x) = 1/(4/5 ⇒ csc(x) = 5/4

Finally sin(x)csc(x)=(4/5)(5/4) ⇒sin(x)csc(x)= 1.