How to find the sine of a missing side - ACT Math

Card 0 of 3

Question

In the triangle below, units and units. What is $\sin C$ ?

Answer

Because $\sin C =\frac{AB}{BC}$ , we need to first find the length of BC.

Using the Pythagorean theorem,

$BC=\sqrt {AB^2+AC^2}$

$BC=\sqrt {6^2+2^2}$

$BC=\sqrt{36+4} = \sqrt {40} =\sqrt{4\cdot10}$

$BC=2\sqrt10$ .

$\sin C =\frac{AB}{BC}=\frac{2}{2\sqrt10}=\frac{1}{\sqrt10}$

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Question

A right triangle has leg lengths and . What is the sine of the angle opposite from the side of length ?

Answer

Using SOHCAHTOA, the sine of an angle is simply the length of the side opposite to it over the hypotenuse; however, we do not have the length of the hypotenuse yet.

Using the Pythagorean Theorem, we can solve for it:

$8^2+10^2=h^2{}$

$h=\sqrt{164}=2\sqrt{41}{}$

So, the sine of this angle is:

$\frac{10}{2\sqrt{41}} = \frac{5}{\sqrt{41}}{}$

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Question

What is the value of in the triangle above? Round to the nearest hundredth.

Answer

We know that the sine of an angle is:

$\frac{opposite}{hypotenuse}$

Therefore, we can write for this question:

$sin(50)=\frac{x}{14}$

This allows us to solve for easily:

Rounding, this gives us .

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